[lars512]

I prefer not to use "god" as a pronoun, since it makes assumptions which aren't warranted. Let's remove the assumptions by using "a god" instead, and look at your sentences again.

"I do not believe a god exists."

"I believe a god does not exist."

They look the same to me.

[babblefrog]

Maybe it's a problem with English. I can say that I don't believe in the existence of life on a planet orbiting Alpha Centauri, but that I don't believe in the non-existence either. In other words I have no data, so how could I believe either one?