There's actually a third way too, though it suffers from somewhat severe lag and image quality limitations (adequate for turn based RPGs or strategy games but a bit hobbling for any real time action games). Namely, if you happen to have a webcam that has s-video input (not as uncommon as you think, mine is just some straight logitech webcam), you can plug the console video output into the webcam (and the webcam into the computer, presumably through USB). You'll have to do the sound separately (just pick up the appropriate converter at Radioshack and you can plug it straight into the audio in on your computer). Then, fire up the console, fire up your webcam application of choice, and there it is.
The end result is actually kind of cool in one way, and that is that you can turn on the actual webcam software and broadcast a continually updating screenshot of whatever you're playing. I did this (with a PS2 not an Xbox but it'd work the same way), and whenever I played I put up an away message saying "I'm playing ps2, visit 220.127.116.11/webcam for a screenshot."
Anyway, not totally practical I realize, but I played through the better part of Xenosaga this way. That, and one of my friends wasted tons of time at his job continually refreshing the screenshot to watch me play.