I have done some research and figured out how to directly download YouTube videos in both high quality (mp4) and normal quality (flv) from the web site with only Internet Explorer, and without visiting any sites other than the YouTube page.
The first site has a really big bookmarklet which is slightly outdated but provides a lot of insight into how the systems of various sites actually work (or did work at the time it was made). You might notice that the bookmarklet is incredibly long; this is because it encodes several images as extra base-64 data, and there’s also a lot of code in there.
The second site has another bookmarklet, which is pretty slick since it acts like a little add-on for the YouTube page, and adds a link below the video information section.
For formatting reasons I have moved the actual details of the method to this page. I’d love to post it here like before, but it was literally breaking WordPress. I guess there was just too much HTML in there and all the formatting got used up for the rest of the page.
Feel free to leave comments if you figure out how to do these any better, for other sites, or if you just want to tell me something.