I think "avatar" originally meant a 3D, at least partially articulated representation on The Net. Think "Snow Crash" or "Neuromancer." That has been extended over the years to include both animated and static 2D representations. The extension to "screen name" is tenuous. At this point, it's probably okay to say that it is an incorrect usage. But it's not clear to me how wide-spread a misusage has to be before one just accepts it.

Are "web" and "net" synonymous? Of course not. But for many people they are. In my mind it's very difficult for me to not consider this an erroneous usage - and yet, for many people, they're the same.