Ultimately everyone has a slightly different definition. To a Japanese person "anime" can be any kind of animation. To an anime fan who isn't into normal animation, it differentiates stuff aimed at the Japanese market from the rest of the medium. And to a more casual fan, all of the "grey area" content such as non-Japanese animation made in Asia, western-commissioned anime or western animation which borrows styles, settings and themes from Japan is usually lumped in with anime as well.
In the case of Avatar I'd personally say it's not anime by my definition as a westerner - it has nothing to do with Japan (aside from one faction in the series who aren't mentioned very often) and from watching it I'd say it heavily borrows from kung fu films far more than from anime aside from the superficial character design similarities. I thought Avatar was tremendously good fun, much better than I was expecting for something which borrowed so heavily, so whether people want to call it anime or not doesn't bother me so much. I think that the US should be proud to call it western animation though (I'm ignoring that it was animated in Korea as most things are lately); it's one of the precious few series I've enjoyed in the last few years from them and there are a lot of western values/style choices in there to complement the nods to Asian traditions.
Sort of scared about the movie but trying to keep an open mind until I see it.
The topic reminds me a little of the annoyance I always privately feel when I am in a comic shop and non-manga - in the strict sense of the word- is mixed in with the manga; I'm not interested in reading romantic Korean comics I've never heard of so it's a distracted when I'm browsing. My husband however doesn't care where it comes from and is much more open to impulse purchases than I am (I can't stand them!), so he likes the types being mixed together to give him more choice. Of course the comic store owners won't know which is which anyway unless they happen to be enthusiasts too and there's very little practical difference between The Wallflower and any similar manhwa if you're just looking for a new series. I suppose it's the nature of fandom to become very particular about things like that.
Edit: I was going to excise this discussion and make it into its own topic as it's interesting to people who don't usually stray into this thread as well, however this thread is so large the forum software seems to choke when I try to split it, sorry!
R