(Didn't read whole thread, aslo I'm not linguist at all. But I am Russian, so I tried to understand difference by myself. Hope, this will be helpfull)
For the most part, `аватАра` и `аватАре` are pronounced similar. You don't need to give attention to it.
But the light difference exists.
For example, if `аватар` means small user's picture on forum, then:
`У тебя нет аватАра` - pronounces with strength in it. Guess, it's something like `fIght`
`Что это у тебя на аватАре?` - pronounces softer than former one. It has nothing to do with exclusions or rules. It's because `е` in the end is "soft" sound, which makes `р` softer, and vice versa. So, `аватАре` gets his softness from `е`.