user8157: I read the following on the internet:

About this unusual understanding of sati ("Sati is never present moment awareness") I hope the reader of this answer put some time
  reading the suttas to come to his/her own conclusion about what sati
  can refer to.

Where is the Palis suttas is 'sati' defined as 'present moment awareness'?

user10100: There are variant "present" words'  meanings in tipitaka and commentaries because of variant sutta contexts such as ability of listener, etc.
Commentaries collected  "present" words in tipitaka as 3 periods: present by rebirth period, present by doing period, and present by citta&rūpa arising period. Another  "present" word that focus on arising khandha, now.
present by birth period (addhāna-paccupanna)
The kammaṭṭhāna teacher, such as buddha, will teach this meaning to the beginner practitioner who has gotten jhāna or uggatitaññū/diṭṭhicarita-vipacitaññū. So this meaning always use in suttanta-pitaka especially paṭiccasamuppāda. So in abhidhamma-pitaka, vibhaṅga that was memorized by sariputta-mahāsāvaka then retaught in his  literal,  wrote that paṭiccasamuppāda by rebirth, is suttanta-method. And in paṭisambhidāmagga, that author by sāriputta-mahāsāvaka, wrote that viññāṇa-paṭiccasamuppāda is rebirth (paṭisaṇdhi). Also bhaddekarattasutta.
present by doing period (santati-paccupanna)
The kammaṭṭhāna teacher, such as buddha, will teach this meaning to a fool who can not  get high level vipassanā (udayabbayañāṇa) or neyya/taṇhācarita-vipacitaññū. For the example in mahāsatipaṭṭhānasutta "sāmissaṃ vā sukkhaṃ vedanaṃ vediyamāno sāmissaṃ sukkhaṃ vedanaṃ vediyāmīti pajānāti". Or in commentary such as arūpadhammānaṃ upaṭṭhānākārakathā 
 and indriyatikkhakāraṇanavakakathā of visuddhimagga.
***Doing period is period between after come from the hotter temperature place then sit in the colder place, between the temperature changing is present doing period. Another same vīthi's javanāni are present doing period.
present by citta&rūpa period  (khaṇa-paccupanna, paccuppannā dhammā)
***Khaṇa is  the smallest period in the universe. It is just arising part+lving part+vanishing part (3 anukhaṇa) of citta&rūpa. There are more than trillion khanā while just snapping fingers. Every suffering truth that have own causes to make it's arising, has their own khaṇa period. So 11 group khandha have their own present arising: past khandha arising, future khandha arising, present khandha arising, self khandha arising, others khandha arising, not smooth khandha arising, smooth khandha arising, good khandha arising, and bad khandha khandha arising. So past and future khandha are reality, that is the object of vipassanā meditation, too.
Arising Present khandha As the Object Of Vipassanā Meditation
The kammaṭṭhāna teacher, such as buddha, will teach this meaning to the advance practitioner who can clearly comprehend paṭiccasamuppāda. Because practitioner who already understand paṭiccasamuppāda, has already seen anicca/dukkha/anatta-lakkhaṇa of the whole khandha (11 group khandha: past, future, present, self, others, not smooth, smooth, good, and bad khandha). He already has auto clearly comprehended in them, so when he is sitting to do vipassanā meditate, without the other jobs, his object should be the present himself that  comprehending another object. Present himself, arising khandha khaṇa, is the closest khandha to 
 be attached by taṇhā/diṭṭhi/māna. The mahāsatipaṭṭhānasutta's commentary very focus in this meaning, because kuru people are uggaṭitaññū.
P.S. It is very very deep,very hard to explain. Please see every  reference to understand  "present".

user11009: Sutta does not define that 'sati' as present moment awareness.
Mindfulness in Buddhist teaching is much more than the english meaning of mindfulness and the the general mindfulness taught in the west.
To understand the meaning of 'Sati' we should have a good understanding of the following two Suttas.
http://www.accesstoinsight.org/tipitaka/mn/mn.010.than.html
http://www.accesstoinsight.org/tipitaka/mn/mn.118.than.html

user-1: Putting aside the fact that pāli->english translators have chosen the word "mindful" (mindfulness, mindfully, etc) to translate sati in many occasions, and that this word is associated with awareness...

Suppose a man would enter a thorny forest. There would be thorns in front of him, thorns behind him, thorns to his left, thorns to his right, thorns below him, thorns above him. He would go forward mindfully (satova), he would go back mindfully (satova), thinking, ‘May no thorn prick me!’
-- SN 35.244

The above passage explains how a person is attentive to the surrounding dangers while moving around. sati is used to characterize his attitude. I suppose one can choose to read this passage as meaning strictly that the words 'May no thorn prick me!' is in that man's mind. However, such a reading doesn't seem very appealing: how having these words in mind would help this person in this dangerous situation?
On another vein, SN 48.11 defines the satindriya  as the result of the satipatthana practice. And we all know that satipatthana is not a mere memory exercise.
While I never believed sati to mean what is usually called "bare awareness", I'd be very cautious about completely divorcing the meaning of sati from awareness (e.g. of what is going on, of what one is doing or of something present in the present moment).
For the reader looking for further information, I suggest looking for translators comments and books dealing with this subject (e.g. "Mindfulness in early buddhism" by Tse-fu Kuan).

user254: Sati is defined in the Sati Sutta (SN 47.35).
Relevant words from that definition seem to me to be viharati and ...passī:

viharati -- stay, abide, dwell, sojourn
...passī -- observing

So, "to dwell observing": that is more-or-less "present moment awareness", isn't it?

Also maybe Thiago meant SN 48.10 rather than SN 48.11. 
The definition there is the same as in SN 47.35, i.e.,

Katamañ-ca, bhikkhave, Satindriyaṁ?
  And what, monks, is the Faculty of Mindfulness?
Idha, bhikkhave, ariyasāvako satimā hoti,
  Here, monks, a noble disciple is mindful,
paramena satinepakkena samannāgato,
  endowed with superior mindfulness and carefulness,
cirakatam-pi cirabhāsitam-pi saritā anussaritā.
  remembering and recalling what was done a long time ago and what was said a long time ago.
So kāye kāyānupassī viharati,
  He dwells contemplating (the nature of) the body in the body,
ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.
Vedanāsu vedanānupassī viharati,
  He dwells contemplating (the nature of) feelings in feelings,
ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.
Citte cittānupassī viharati,
  He dwells contemplating (the nature of) the mind in the mind,
ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.
Dhammesu dhammānupassī viharati,
  He dwells contemplating (the nature of) things in (various) things
ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.
Idaṁ vuccati, bhikkhave, satindriyaṁ.
  This, monks, is called the Faculty of Mindfulness.

... except that it also introduces two other concepts at the beginning, i.e. "carefulness" and "remembering and recalling".
Bikkhu Bodhi said,

Even the word sati, rendered mindfulness, isn’t unproblematic. The word derives from a verb, sarati, meaning “to remember,” and occasionally in Pali sati is still explained in a way that connects it with the idea of memory. But when it is used in relation to meditation practice, we have no word in English that precisely captures what it refers to. An early translator cleverly drew upon the word mindfulness, which is not even in my dictionary. This has served its role admirably, but it does not preserve the connection with memory, sometimes needed to make sense of a passage.

... so it implies "remembering" too

8157: <p>I read the following on the internet:</p>

<blockquote>
  <p><em>About this unusual understanding of sati ("Sati is never present moment awareness") I hope the reader of this answer put some time
  reading the suttas to come to his/her own conclusion about what sati
  can refer to.</em></p>
</blockquote>

<p>Where is the Palis suttas is 'sati' defined as 'present moment awareness'?</p>
 10100: <p>There are variant "present" words'  meanings in tipitaka and commentaries because of variant sutta contexts such as ability of listener, etc.</p>

<p>Commentaries collected  "present" words in tipitaka as 3 periods: present by rebirth period, present by doing period, and present by citta&amp;rūpa arising period. Another  "present" word that focus on arising khandha, now.</p>

<h2>present by birth period (addhāna-paccupanna)</h2>

<p>The kammaṭṭhāna teacher, such as buddha, will teach this meaning to the beginner practitioner who has gotten jhāna or uggatitaññū/diṭṭhicarita-vipacitaññū. So this meaning always use in suttanta-pitaka especially paṭiccasamuppāda. So in abhidhamma-pitaka, vibhaṅga that was memorized by sariputta-mahāsāvaka then retaught in his  literal,  wrote that paṭiccasamuppāda by rebirth, is suttanta-method. And in paṭisambhidāmagga, that author by sāriputta-mahāsāvaka, wrote that viññāṇa-paṭiccasamuppāda is rebirth (paṭisaṇdhi). Also <strong>bhaddekarattasutta</strong>.</p>

<h2>present by doing period (santati-paccupanna)</h2>

<p>The kammaṭṭhāna teacher, such as buddha, will teach this meaning to a fool who can not  get high level vipassanā (udayabbayañāṇa) or neyya/taṇhācarita-vipacitaññū. For the example in mahāsatipaṭṭhānasutta "sāmissaṃ vā sukkhaṃ vedanaṃ vediyamāno sāmissaṃ sukkhaṃ vedanaṃ <strong>vediyāmīti</strong> pajānāti". Or in commentary such as <strong>arūpadhammānaṃ upaṭṭhānākārakathā</strong> 
 and <strong>indriyatikkhakāraṇanavakakathā</strong> of visuddhimagga.</p>

<p>***Doing period is period between after come from the hotter temperature place then sit in the colder place, between the temperature changing is present doing period. Another same vīthi's javanāni are present doing period.</p>

<h2>present by citta&amp;rūpa period  (khaṇa-paccupanna, paccuppannā dhammā)</h2>

<p>***Khaṇa is  the smallest period in the universe. It is just arising part+lving part+vanishing part (3 anukhaṇa) of citta&amp;rūpa. <strong>There are more than trillion khanā while just snapping fingers.</strong> Every suffering truth that have own causes to make it's arising, has their own khaṇa period. So 11 group khandha have their own present arising: past khandha arising, future khandha arising, present khandha arising, self khandha arising, others khandha arising, not smooth khandha arising, smooth khandha arising, good khandha arising, and bad khandha khandha arising. <strong>So past and future khandha are reality, that is the object of vipassanā meditation, too.</strong></p>

<h2>Arising Present khandha As the Object Of Vipassanā Meditation</h2>

<p>The kammaṭṭhāna teacher, such as buddha, will teach this meaning to the advance practitioner who can clearly comprehend paṭiccasamuppāda. Because practitioner who already understand paṭiccasamuppāda, has already seen anicca/dukkha/anatta-lakkhaṇa of the whole khandha (11 group khandha: past, future, present, self, others, not smooth, smooth, good, and bad khandha). He already has auto clearly comprehended in them, so when he is sitting to do vipassanā meditate, without the other jobs, his object should be the present himself that  comprehending another object. Present himself, arising khandha khaṇa, is the <strong>closest khandha to 
 be attached by taṇhā/diṭṭhi/māna.</strong> The mahāsatipaṭṭhānasutta's commentary <strong>very focus in this meaning, because kuru people are uggaṭitaññū.</strong></p>

<p>P.S. It is very very deep,very hard to explain. Please see every  reference to understand  "present".</p>
 11009: <p>Sutta does not define that 'sati' as present moment awareness.
Mindfulness in Buddhist teaching is much more than the english meaning of mindfulness and the the general mindfulness taught in the west.
To understand the meaning of 'Sati' we should have a good understanding of the following two Suttas.</p>

<p><a href="http://www.accesstoinsight.org/tipitaka/mn/mn.010.than.html" rel="nofollow noreferrer">http://www.accesstoinsight.org/tipitaka/mn/mn.010.than.html</a></p>

<p><a href="http://www.accesstoinsight.org/tipitaka/mn/mn.118.than.html" rel="nofollow noreferrer">http://www.accesstoinsight.org/tipitaka/mn/mn.118.than.html</a></p>
 -1: <p>Putting aside the fact that pāli->english translators have chosen the word "mindful" (mindfulness, mindfully, etc) to translate <em>sati</em> in many occasions, and that this word is associated with awareness...</p>

<blockquote>
  <p>Suppose a man would enter a thorny forest. There would be thorns in front of him, thorns behind him, thorns to his left, thorns to his right, thorns below him, thorns above him. He would go forward mindfully (<em>satova</em>), he would go back mindfully (<em>satova</em>), thinking, ‘May no thorn prick me!’</p>
  
  <p>-- SN 35.244</p>
</blockquote>

<p>The above passage explains how a person is attentive to the surrounding dangers while moving around. <em>sati</em> is used to characterize his attitude. I suppose one can choose to read this passage as meaning strictly that the words 'May no thorn prick me!' is in that man's mind. However, such a reading doesn't seem very appealing: how having these words in mind would help this person in this dangerous situation?</p>

<p>On another vein, SN 48.11 defines the <em>satindriya</em>  as the result of the satipatthana practice. And we all know that satipatthana is not a mere memory exercise.</p>

<p>While I never believed <em>sati</em> to mean what is usually called "bare awareness", I'd be very cautious about completely divorcing the meaning of <em>sati</em> from awareness (e.g. of what is going on, of what one is doing or of something present in the present moment).</p>

<p>For the reader looking for further information, I suggest looking for translators comments and books dealing with this subject (e.g. "Mindfulness in early buddhism" by Tse-fu Kuan).</p>
 254: <p><em>Sati</em> is defined in the <a href="http://www.buddha-vacana.org/sutta/samyutta/maha/sn47-035.html" rel="nofollow noreferrer">Sati Sutta (SN 47.35)</a>.</p>

<p>Relevant words from that definition seem to me to be <em>viharati</em> and <em>...passī</em>:</p>

<ul>
<li><em>viharati</em> -- stay, abide, dwell, sojourn</li>
<li><em>...passī</em> -- observing</li>
</ul>

<p>So, "to dwell observing": that is more-or-less "present moment awareness", isn't it?</p>

<hr>

<p>Also maybe Thiago meant <a href="https://www.ancient-buddhist-texts.net/Texts-and-Translations/Short-Pieces/Indriyavibhangasuttam.htm" rel="nofollow noreferrer">SN 48.10</a> rather than SN 48.11. </p>

<p>The definition there is the same as in SN 47.35, i.e.,</p>

<blockquote>
  <p>Katamañ-ca, bhikkhave, Satindriyaṁ?<br>
  And what, monks, is the Faculty of Mindfulness?</p>
  
  <p>Idha, bhikkhave, ariyasāvako satimā hoti,<br>
  Here, monks, a noble disciple is mindful,</p>
  
  <p>paramena satinepakkena samannāgato,<br>
  endowed with superior mindfulness and carefulness,</p>
  
  <p>cirakatam-pi cirabhāsitam-pi saritā anussaritā.<br>
  remembering and recalling what was done a long time ago and what was said a long time ago.</p>
  
  <p>So kāye kāyānupassī viharati,<br>
  He dwells contemplating (the nature of) the body in the body,</p>
  
  <p>ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.<br>
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.</p>
  
  <p>Vedanāsu vedanānupassī viharati,<br>
  He dwells contemplating (the nature of) feelings in feelings,</p>
  
  <p>ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.<br>
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.</p>
  
  <p>Citte cittānupassī viharati,<br>
  He dwells contemplating (the nature of) the mind in the mind,</p>
  
  <p>ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.<br>
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.</p>
  
  <p>Dhammesu dhammānupassī viharati,<br>
  He dwells contemplating (the nature of) things in (various) things</p>
  
  <p>ātāpī, sampajāno, satimā, vineyya loke abhijjhādomanassaṁ.<br>
  ardent, clearly knowing, and mindful, after removing avarice and sorrow regarding the world.</p>
  
  <p>Idaṁ vuccati, bhikkhave, satindriyaṁ.<br>
  This, monks, is called the Faculty of Mindfulness.</p>
</blockquote>

<p>... except that it also introduces two other concepts at the beginning, i.e. "carefulness" and "remembering and recalling".</p>

<p><a href="http://www.inquiringmind.com/Articles/Translator.html" rel="nofollow noreferrer">Bikkhu Bodhi said</a>,</p>

<blockquote>
  <p>Even the word sati, rendered mindfulness, isn’t unproblematic. The word derives from a verb, sarati, meaning “to remember,” and occasionally in Pali sati is still explained in a way that connects it with the idea of memory. But when it is used in relation to meditation practice, we have no word in English that precisely captures what it refers to. An early translator cleverly drew upon the word mindfulness, which is not even in my dictionary. This has served its role admirably, but it does not preserve the connection with memory, sometimes needed to make sense of a passage.</p>
</blockquote>

<p>... so it implies "remembering" too</p>
