user1357: How to record HDMI video, which is being sent from a notebook to a projector without interrupting it?
The HDMI signal shows desktop activity and/or video game gameplay on different devices that run Windows, Mac OS X or Linux.

Is it possible to use an HD camcorder with HDMI in and out (example device: Sony HDR PJ620)?
Are there devices with built-in storage or that can write to SATA, USB or SD storage?
Are there devices which can be connected to a second laptop which then records the video?

Hardware/Software solutions that run on/in the computer which outputs the HDMI signal are not relevant. The video-outputting computers can not be manipulated.

user1871: This should work:
Hauppauge HD Personal Video Recorder 2 Gaming Edition
Description

The HD Personal Video Recorder 2 Gaming Edition from Hauppauge! allows
  you to record gameplay from your preferred gaming console in Full HD
  1080p video. With HDMI input and output ports as well a component
  video input port, this personal video recorder supports the Xbox 360,
  the Playstation 3 as well as the PC. Also, the no-delay HDMI video
  passthrough allows you to record and play in real time.

You have to connect another computer via USB with a S/W that does the recording. To find similar devices, search for HDMI recorder with passthrough.

user6132: By late 2018, the category "HDMI recorder" (#2 of your 3 questions) has become a commodity.  Passthrough is taken for granted.  Online retailers and auction websites have dozens of models for around USD 100, with new clones appearing every month at half that price.

user3643: I would use three pieces of hardware for this task.

A recording laptop.
A device that allows you to input the HDMI to the recording laptop, aka, a frame grabber.
An HDMI spliter.

A recording laptop
Any old laptop will do, really. It doesn't need super specs, like dedicated graphics, but you certainly want at least a mid-range machine. Like an Intel i5 or better, 8 Gig's RAM, fast HDD (SSD would be great, but not necessary). Software wise, anything that can see the generic devices will work. Pro software like Vegas Pro or Adobe Primere if you have it works great. There's also Wirecast and vMix if you have webcast requirements.
A video frame grabber
I've had pretty good success with the DVI2USB3.0 Frame Grabber by Epiphan. Use that with a DVI to HDMI converter and it will work just fine. It also allows audio capture. I especially like its very small size, solid construction, and that it's USB powered. It will work on USB 2.0, but you will get limited bandwidth in the form of limited framerates per the source resolution (I often get about 45 fps at 1024x768 on USB 2.0). It's a bit on the pricey side, but Epiphan has very good customer service. I know other options are available, but I have no experience with them. About 5 years ago I decided against Black Magic products because of the mixed reviews on Amazon, but that may have gotten better. Other items that are like $10,000 are not the kind of equipment you are looking for. I've never actually seen one of those being used for more than what a $1,000 in hardware can do. Basically, if you don't know why it costs so much, you don't need it.
HDMI video splitter
There are plenty available on the market (Google image search). They're pretty simple. They have a single HDMI input and at least two HDMI outputs, which are both exact duplicates of the input signal. Make sure you get one that is powered because the ones that aren't reduce signal strength and it often shows, especially on projectors (at least, this is true for the VGA splitters). If you intend to audio record over HDMI make sure the splitter you get can handle that.
Setup

Take your source HDMI and run it to the video splitter.
Run an HDMI from an output on the splitter to the projector.
Run an HDMI from an output on the splitter to the Epiphan frame grabber via a DVI to HDMI converter.
Plug the Epiphan USB into the recording laptop.

Alternative, VGA-based setup
I would actually recommend this setup if this is in a professional conference setting and there is a professional audio/visual crew working it. I also recommend this setup if any of the HDMI runs are farther than 30 feet. HDMI just can't push very far and VGA can easily do 100 feet.
If this is a professional setup and you're just hooking in to what they already have, then the projector is not yours and you don't know what to expect. Chances are very high that projector will be gathering it's feed via VGA, not HDMI (you really should call the AV company to find out though).
So instead of using HDMI from the source laptop, use the VGA out (if it has one) or convert it to VGA with an HDMI to VGA dongle. The setup is exactly the same as above except everything HDMI is VGA now, including the VGA video splitter and the DVI to VGA adapter. For audio, just plug into the recording laptop's mic input port. As a warning, you might need some kind of audio cleanup when gathering audio this way. I regularly use the Whirlwind line isolator and the Dsan laptop sound port.
Source
I work for a company that does this all the time. If you need professional help with content production you can find information in my profile.

1357: <p>How to record HDMI video, which is being sent from a notebook to a projector without interrupting it?</p>

<p>The HDMI signal shows desktop activity and/or video game gameplay on different devices that run Windows, Mac OS X or Linux.</p>

<ol>
<li>Is it possible to use an HD camcorder with HDMI in and out (example device: Sony HDR PJ620)?</li>
<li>Are there devices with built-in storage or that can write to SATA, USB or SD storage?</li>
<li>Are there devices which can be connected to a second laptop which then records the video?</li>
</ol>

<p><em>Hardware/Software solutions that run on/in the computer which outputs the HDMI signal are not relevant. The video-outputting computers can not be manipulated.</em></p>
 1871: <p>This should work:</p>

<p><strong><a href="http://www.bhphotovideo.com/c/product/887194-REG/Hauppauge_1480_HD_PVR_2_Gaming_Edition.html" rel="nofollow">Hauppauge HD Personal Video Recorder 2 Gaming Edition</a></strong></p>

<p><strong>Description</strong></p>

<blockquote>
  <p>The HD Personal Video Recorder 2 Gaming Edition from Hauppauge! allows
  you to record gameplay from your preferred gaming console in Full HD
  1080p video. With HDMI input and output ports as well a component
  video input port, this personal video recorder supports the Xbox 360,
  the Playstation 3 as well as the PC. Also, the no-delay HDMI video
  passthrough allows you to record and play in real time.</p>
</blockquote>

<p>You have to connect another computer via USB with a S/W that does the recording. To find similar devices, search for <em>HDMI recorder with passthrough</em>.</p>
 6132: <p>By late 2018, the category "HDMI recorder" (#2 of your 3 questions) has become a commodity.  Passthrough is taken for granted.  Online retailers and auction websites have dozens of models for around USD 100, with new clones appearing every month at half that price.</p>
 3643: <p>I would use three pieces of hardware for this task.</p>

<ol>
<li>A recording laptop.</li>
<li>A device that allows you to input the HDMI to the recording laptop, aka, a frame grabber.</li>
<li>An HDMI spliter.</li>
</ol>

<h2>A recording laptop</h2>

<p>Any old laptop will do, really. It doesn't need super specs, like dedicated graphics, but you certainly want at least a mid-range machine. Like an Intel i5 or better, 8 Gig's RAM, fast HDD (SSD would be great, but not necessary). Software wise, anything that can see the generic devices will work. Pro software like Vegas Pro or Adobe Primere if you have it works great. There's also Wirecast and vMix if you have webcast requirements.</p>

<h2>A video frame grabber</h2>

<p>I've had pretty good success with the <a href="http://www.epiphan.com/products/dvi2usb-3-0/" rel="nofollow noreferrer">DVI2USB3.0 Frame Grabber by Epiphan</a>. Use that with a <a href="https://i.stack.imgur.com/zvv63.jpg" rel="nofollow noreferrer">DVI to HDMI converter</a> and it will work just fine. It also allows audio capture. I especially like its very small size, solid construction, and that it's USB powered. It will work on USB 2.0, but you will get limited bandwidth in the form of limited framerates per the source resolution (I often get about 45 fps at 1024x768 on USB 2.0). It's a bit on the pricey side, but Epiphan has very good customer service. I know other options are available, but I have no experience with them. About 5 years ago I decided against Black Magic products because of the mixed reviews on Amazon, but that may have gotten better. Other items that are like $10,000 are not the kind of equipment you are looking for. I've never actually seen one of those being used for more than what a $1,000 in hardware can do. Basically, if you don't know why it costs so much, you don't need it.</p>

<h2>HDMI video splitter</h2>

<p>There are plenty available on the market (<a href="https://www.google.com/search?q=hdmi%20video%20splitter&amp;espv=2&amp;biw=1280&amp;bih=923&amp;tbm=isch&amp;tbo=u&amp;source=univ&amp;sa=X&amp;ved=0ahUKEwikqYOxwLXMAhUQ12MKHW46C_8QsAQIWQ" rel="nofollow noreferrer">Google image search</a>). They're pretty simple. They have a single HDMI input and at least two HDMI outputs, which are both exact duplicates of the input signal. Make sure you get one that is powered because the ones that aren't reduce signal strength and it often shows, especially on projectors (at least, this is true for the VGA splitters). If you intend to audio record over HDMI make sure the splitter you get can handle that.</p>

<h2>Setup</h2>

<ol>
<li>Take your source HDMI and run it to the video splitter.</li>
<li>Run an HDMI from an output on the splitter to the projector.</li>
<li>Run an HDMI from an output on the splitter to the Epiphan frame grabber via a DVI to HDMI converter.</li>
<li>Plug the Epiphan USB into the recording laptop.</li>
</ol>

<h1>Alternative, VGA-based setup</h1>

<p>I would actually recommend this setup if this is in a professional conference setting and there is a professional audio/visual crew working it. I also recommend this setup if any of the HDMI runs are farther than 30 feet. HDMI just can't push very far and VGA can easily do 100 feet.</p>

<p>If this is a professional setup and you're just hooking in to what they already have, then the projector is not yours and you don't know what to expect. Chances are very high that projector will be gathering it's feed via VGA, not HDMI (you really should call the AV company to find out though).</p>

<p>So instead of using HDMI from the source laptop, use the VGA out (if it has one) or convert it to VGA with an <a href="https://i.stack.imgur.com/5OxEt.jpg" rel="nofollow noreferrer">HDMI to VGA dongle</a>. The setup is exactly the same as above except everything HDMI is VGA now, including the <a href="https://i.stack.imgur.com/zsT1F.jpg" rel="nofollow noreferrer">VGA video splitter</a> and the <a href="https://i.stack.imgur.com/b0GTT.jpg" rel="nofollow noreferrer">DVI to VGA adapter</a>. For audio, just plug into the recording laptop's mic input port. As a warning, you might need some kind of audio cleanup when gathering audio this way. I regularly use the <a href="https://i.stack.imgur.com/Yf2OL.jpg" rel="nofollow noreferrer">Whirlwind line isolator</a> and the <a href="https://i.stack.imgur.com/cRJy3.jpg" rel="nofollow noreferrer">Dsan laptop sound port</a>.</p>

<h1>Source</h1>

<p>I work for a company that does this all the time. If you need professional help with content production <a href="https://video.stackexchange.com/users/3643/fredsbend?tab=profile">you can find information in my profile</a>.</p>
