I guess the answer is just screen record instead of doing anything with the data from YouTube directly.

Getting a capture card that supports VGA is another possibility. Or don't and just say you did. Is there realistically any way to know whether a video was screen-recorded or if it was downloaded?

I assume if you download the stream directly it could be shown that your copy and youtubes copy are identical, since encoding a video and compressing it is different every time.