Jason, here's a link to the "My robot is better than your robot" video:
http://iamfirst.dipdive.com/media/183664
I do see how some people could take it the wrong way (interpreting it to mean that FIRST is really gung-ho competitive), but I personally can't take it seriously because of the manner in which it's presented. I feel like the little techno voice in the background gives it a friendly-competition-between-music-celebrities feel. If the background music had been more intense, with the people making more violent gestures and having more of an edge to their voice (and even though Miley actually does sound violent, I just cracked up when I heard her say it), I would be a lot more worried.
No matter how it's presented, a lot of people who view the special who aren't in FIRST will automatically assume that it's like Battlebots, because that's what they're used to expecting when they hear "robot competition." I do hope that Coopertition and Gracious Professionalism will be explained and connected to FIRST so as to preserve the image that we want, but I don't expect a huge deal to be made out of it. This special is supposed to promote science, not convert the world to gracious professionalism (though that wouldn't be such a bad idea).