user12009: What is the Sanskrit term for a person who is a follower of the Buddha, i.e. a "Buddhist"?
Things I tried: I searched online but couldn't find anything. I looked in the only Sanskrit I have access to, which is The Student's Sanskrit English Dictionary by Vaman Shivram Apte. It has a term in it, but the Devanagari is so small, I can't make out the letters. It looks like the term is "buddhupasaka" or "buddhupasakra", and it is defined as "a worshipper of Buddha". 

user43: I'm not a Sanskrit scholar but a quick dictionary search gives the following:
bhikṣu - a beggar/wanderer
śramaṇa - a spiritual seeker
pravrājita - the one gone forth
anusārī - follower  
The problem is, back in Buddha's times there were hundreds of different sects, and every seeker either went from one sect to another and assembled their own dharma from the elements available on the spiritual market - or picked one teacher to follow. In the first case they were simply called a bhikshu or shramana, and in the second case they were called a follower of such and such teacher (e.g. of Shramana Gautama, "sramanagautamanusari").
So the closest to "Buddhist" would probably be "buddha-pravrājita" or "buddha-anusārī"

user471: The term Dharmānusārin would mean "one who follows in the Dharma" in Sanskrit. Please see the definition in this dictionary entry.
The term Śraddhānusārin would mean "one who follows in faith" in Sanskrit. Please see the definition in this dictionary entry.
The words Dharma and Śraddhā can be merged with anusārin according to sandhi rules.
On this page, from S.N. Goenka's Keynote address at the International Bauddha Mahotsav (Buddhist Conference), at Sarnath, India in 1998:

If I had been aware, much earlier, that this conference was named as
  'Bauddha Mahotsav', then I would have suggested that a more
  appropriate name would be 'Buddha Mahotsav', or 'Dhamma Mahotsav', or
  even 'Tiratana Mahotsav'. This is because in the entire ancient
  literature of the Buddha's teachings, commentaries and
  sub-commentaries, amongst a total of 59,150 pages containing 9,285,755
  words, the word 'Bauddha' is very conspicuously absent. The Buddha
  never taught 'Buddhism' or 'Bauddh Dharma'. He never taught a
  religion, nor did he convert anyone to an organized religion. He
  taught Dhamma (Dharma) - the universal laws of nature - and inspired
  people to follow Dhamma.
Those who followed the teachings of the Buddha were never called
  'Bauddh' (Buddhist) during the time of the Buddha, and even till about
  500 years later. Throughout this huge literature, we find only the
  following words referring to those practicing the Buddha's teachings:
  Dhammi (Dharmi), Dhammiko (Dharmika), Dhammattho (Dharmastha),
  Dhammacari (Dharmacari), Dhammavihari (Dharmavihari), Dhammanusari
  (Dharmanusari).
We don't know when, where, by whom and why the use of the word
  'Bauddh' first began, and so also the use of the words 'Buddhism' and
  'Buddhist'. When we say that the teachings of the Buddha are 'Bauddh
  Dharama' (or Buddhism), and say his followers are Bauddh (or
  Buddhists), obviously it means that the teachings are meant only for
  people claiming to be 'Bauddha' or 'Buddhists'. Whereas the Buddha
  declared so emphatically that Dhamma is infinite - "appamano dhammo".
When the teachings are called Buddhist, or ‘Bauddh Dharma' or
  Buddhism, then most of the suffering people of the world will get
  frightened thinking that they are being converted from one particular
  religion to another, and be deprived of the universal teaching. But
  when Buddha's teachings are given under the original, true
  nomenclature of ‘Dhamma', and not any other terminology, then people
  are reassured of the basic fact that the Buddha's teachings are not
  just meant for any one religion. So people from any background will
  have no hesitation accepting his teachings.
Our research at Igatpuri has shown us that till about 500 years after
  Buddha, the word ‘Bauddh' was not found in any ancient spiritual
  literature of India-the literature in the Buddha tradition, Mahavir
  tradition, or Vedic tradition. The distinguished scholars assembled
  here may help us make a proper research as to when usage of this word
  ‘Bauddh' began.
It is most important to remember that the Buddha's teachings are not
  meant to be limited to any particular sect. His teachings are
  universal. And if we see the Enlightened One's teachings in proper
  perspective, it is clear that the original teachings of the Buddha are
  totally universal and non-sectarian.


user10100: In pāli canon, follower is sāvaka (su+ika=follower to hear dhamma from satthā [saha+attha=philosopher who has philosophy in his mind]). 
Sāvaka, in pāli, is śrāvaka, in sanskrit.
This word use in every canon. You can put "buddha" word in font of "sāvaka", buddha-sāvaka, to specify just a buddhist person.
https://www.buddhistdoor.org/tc/dictionary/details/savaka
P.s. use this tool to convert between thai and roman of the first reference link.

user12010: The Sanskrit term for Buddhist is bauddha. 
This derivation is known as a "secondary derivation" (taddhita-pratyaya). An -a suffix is added, and the initial vowel is strengthened. It has the general sense of "of or relating to", and here the specific nuance of "one who follows the Buddha". Compare the more familiar word Jaina, which is the name for a follower of the "Jina", i.e. Mahavira. The word brāhmaṇa is derived from brahman via the same process. 

12009: <p>What is the Sanskrit term for a person who is a follower of the Buddha, i.e. a "Buddhist"?</p>

<p>Things I tried: I searched online but couldn't find anything. I looked in the only Sanskrit I have access to, which is The Student's Sanskrit English Dictionary by Vaman Shivram Apte. It has a term in it, but the Devanagari is so small, I can't make out the letters. It looks like the term is "buddhupasaka" or "buddhupasakra", and it is defined as "a worshipper of Buddha". </p>
 43: <p>I'm not a Sanskrit scholar but a quick dictionary search gives the following:</p>

<p>bhikṣu - a beggar/wanderer<br>
śramaṇa - a spiritual seeker<br>
pravrājita - the one gone forth<br>
anusārī - follower  </p>

<p>The problem is, back in Buddha's times there were hundreds of different sects, and every seeker either went from one sect to another and assembled their own dharma from the elements available on the spiritual market - or picked one teacher to follow. In the first case they were simply called a bhikshu or shramana, and in the second case they were called a follower of such and such teacher (e.g. of Shramana Gautama, "sramanagautamanusari").</p>

<p>So the closest to "Buddhist" would probably be "buddha-pravrājita" or "buddha-anusārī"</p>
 471: <p>The term Dharmānusārin would mean "one who follows in the Dharma" in Sanskrit. Please see the definition in <a href="https://books.google.com.my/books?id=985a1M7L1NcC&amp;lpg=PA74&amp;dq=dharma-anusarin&amp;pg=PA74#v=onepage&amp;q=dharma-anusarin&amp;f=false" rel="nofollow noreferrer">this dictionary entry</a>.</p>

<p>The term Śraddhānusārin would mean "one who follows in faith" in Sanskrit. Please see the definition in <a href="https://books.google.com.my/books?id=985a1M7L1NcC&amp;lpg=PA277&amp;dq=sraddha-anusarin&amp;pg=PA277#v=onepage&amp;q=sraddha-anusarin&amp;f=false" rel="nofollow noreferrer">this dictionary entry</a>.</p>

<p>The words Dharma and Śraddhā can be merged with anusārin according to <a href="http://learnsanskrit.org/references/sandhi/vowel#similar" rel="nofollow noreferrer">sandhi rules</a>.</p>

<p>On <a href="http://www.vridhamma.org/Peace-within-Oneself-for-Peace-in-the-World" rel="nofollow noreferrer">this page</a>, from <a href="https://en.wikipedia.org/wiki/S._N._Goenka" rel="nofollow noreferrer">S.N. Goenka</a>'s Keynote address at the International Bauddha Mahotsav (Buddhist Conference), at Sarnath, India in 1998:</p>

<blockquote>
  <p>If I had been aware, much earlier, that this conference was named as
  'Bauddha Mahotsav', then I would have suggested that a more
  appropriate name would be 'Buddha Mahotsav', or 'Dhamma Mahotsav', or
  even 'Tiratana Mahotsav'. This is because in the entire ancient
  literature of the Buddha's teachings, commentaries and
  sub-commentaries, amongst a total of 59,150 pages containing 9,285,755
  words, the word 'Bauddha' is very conspicuously absent. The Buddha
  never taught 'Buddhism' or 'Bauddh Dharma'. He never taught a
  religion, nor did he convert anyone to an organized religion. He
  taught Dhamma (Dharma) - the universal laws of nature - and inspired
  people to follow Dhamma.</p>
  
  <p>Those who followed the teachings of the Buddha were never called
  'Bauddh' (Buddhist) during the time of the Buddha, and even till about
  500 years later. <strong>Throughout this huge literature, we find only the
  following words referring to those practicing the Buddha's teachings:
  Dhammi (Dharmi), Dhammiko (Dharmika), Dhammattho (Dharmastha),
  Dhammacari (Dharmacari), Dhammavihari (Dharmavihari), Dhammanusari
  (Dharmanusari).</strong></p>
  
  <p>We don't know when, where, by whom and why the use of the word
  'Bauddh' first began, and so also the use of the words 'Buddhism' and
  'Buddhist'. When we say that the teachings of the Buddha are 'Bauddh
  Dharama' (or Buddhism), and say his followers are Bauddh (or
  Buddhists), obviously it means that the teachings are meant only for
  people claiming to be 'Bauddha' or 'Buddhists'. Whereas the Buddha
  declared so emphatically that Dhamma is infinite - "appamano dhammo".</p>
  
  <p>When the teachings are called Buddhist, or ‘Bauddh Dharma' or
  Buddhism, then most of the suffering people of the world will get
  frightened thinking that they are being converted from one particular
  religion to another, and be deprived of the universal teaching. But
  when Buddha's teachings are given under the original, true
  nomenclature of ‘Dhamma', and not any other terminology, then people
  are reassured of the basic fact that the Buddha's teachings are not
  just meant for any one religion. So people from any background will
  have no hesitation accepting his teachings.</p>
  
  <p>Our research at Igatpuri has shown us that till about 500 years after
  Buddha, the word ‘Bauddh' was not found in any ancient spiritual
  literature of India-the literature in the Buddha tradition, Mahavir
  tradition, or Vedic tradition. The distinguished scholars assembled
  here may help us make a proper research as to when usage of this word
  ‘Bauddh' began.</p>
  
  <p>It is most important to remember that the Buddha's teachings are not
  meant to be limited to any particular sect. His teachings are
  universal. And if we see the Enlightened One's teachings in proper
  perspective, it is clear that the original teachings of the Buddha are
  totally universal and non-sectarian.</p>
</blockquote>
 10100: <p><a href="http://www.84000.org/tipitaka/pitaka_item/pali_seek.php?text=%CA%D2%C7%A1&amp;book=1&amp;bookZ=45" rel="nofollow noreferrer">In pāli canon</a>, follower is sāvaka (su+ika=follower to hear dhamma from satthā [saha+attha=philosopher who has philosophy in his mind]). </p>

<p>Sāvaka, in pāli, is <strong>śrāvaka</strong>, in sanskrit.</p>

<p>This word use in every canon. You can put "buddha" word in font of "sāvaka", buddha-sāvaka, to specify just a buddhist person.</p>

<p><a href="https://www.buddhistdoor.org/tc/dictionary/details/savaka" rel="nofollow noreferrer">https://www.buddhistdoor.org/tc/dictionary/details/savaka</a></p>

<p>P.s. use <a href="http://ballwarapol.github.io/pali-editor/paliThaiRoman.html" rel="nofollow noreferrer">this tool</a> to convert between thai and roman of the first reference link.</p>
 12010: <p>The Sanskrit term for Buddhist is <em>bauddha</em>. </p>

<p>This derivation is known as a "secondary derivation" (<em>taddhita-pratyaya</em>). An <em>-a</em> suffix is added, and the initial vowel is strengthened. It has the general sense of "of or relating to", and here the specific nuance of "one who follows the Buddha". Compare the more familiar word Jaina, which is the name for a follower of the "Jina", i.e. Mahavira. The word <em>brāhmaṇa</em> is derived from <em>brahman</em> via the same process. </p>
