Avatar isn't considered Anime in most circles, I guess it falls under this category:
http://en.wikipedia.org/wiki/Anime-influenced_animationI know a friend who made an AMV using Avatar wasn't allowed to enter it into a few anime convention contests because it wasn't considered anime (although some others did allow it). The thing is that Anime isn't an actual dictionary term, it's a term that arose over time and become common use, so different people can believe it to refer to different things.
In general it refers to products made in Japan, for those coming from South Korea, thailand etc they probably still call them anime simply because there is no other term that describes them better, note that comics that come from south Korea are called Manhwa rather than Manga, so different terms for different counties do exist in some cases.