- An atheist believes that God does not exist.
- An atheist does not believe that God exists.
I can't say I object very much to applying the second definition to myself. My main objection is that it seems to be biased toward the belief that God doesn't exist while I'm saying I simply don't know. Actually, I'm pretty sure God doesn't exist as portrayed by religions I know about. There, "does not believe" is entirely appropriate. It just does not seem to be appropriate considering the idea of God in general. (Note that my idea of what the possible nature of God might be is quite broad. God might even be something I cannot imagine.)
Then there's the issue of a term having multiple meanings. I wouldn't want to say I am an atheist using meaning two and then have it misinterpreted as meaning one.
EDIT: Oh, and what about religion which doesn't involve God or deities? Neither definition addresses belief in such religions, and such religions do exist. The core of Buddhism doesn't seem to include deities.