There was discussion earlier in this thread (posts 461, 466-67, 471) about external video cards and recording a computer screen. With AMD cards, the main options are OBS (Open Broadcast Software) and AMD ReLive*.
The advantage of ReLive over OBS is that it should significantly reduce the burden on one's CPU of making the recording. This is important if you want to record something that is CPU intensive, such as a game. However, macOS doesn't support ReLive, so to use it one needs to have Windows on one's Mac.
I have installed Boot Camp Windows on my mini and I know anecdotally that ReLive can be installed and made to work. I'll get there, but installing the current AMD driver package, which includes ReLive, is turning out not to be a simple plug and play exercise
Further to the earlier posts, I can now confirm that there is another solution, one that doesn't require Windows and that
is plug and play. Like ReLive, it takes the burden of recording a computer screen off one's CPU. This involves using a hardware video recorder, which is placed between the external video card and the monitor. These are the connections:
Mac mini via Thunderbolt 3 cable to eGPU enclosure
GPU HDMI port to video recorder "HDMI In" port
Video recorder "HDMI Out" port to monitor HDMI port
With this setup, the video that is sent from my mini (for example, my desktop or a game) appears on both my video recorder screen and my monitor. It worked immediately, requiring no tweaking/fussing. I can record the mini's video output, up to and including 4K, in H.264 or Apple ProRes. ReLive records only in H.264/H.265. However, I need to investigate potential issues. For example, the recorder takes its frame rate from the source, which may or may not be problematic with games.
My recorder is a different brand, but similar to the one described in this recent YouTube review. These recorders are expensive (the one in the review costs US$700), but the idea may be of interest to people who already own, or are thinking of purchasing, an external recorder to use with their camera. The "HDMI In" and "HDMI Out" connections that I referred to above are shown/explained at 02:17:
Right now, I don't know whether this will work better than ReLive, worse or the same, but for me it's an interesting experiment.
* There is a lot of confusion on the internet about how to pronounce ReLive. AMD, in its videos, pronounces it as a verb, as in "She wants to relive the game".