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.
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?
"I'm asking you to not join the Navy."
"I'm not asking you to join the Navy."
The difference in word order is the same as the different phrasings between different atheists:
"I do not believe god exists." (I call this atheism)
"I believe god does not exist." (I call this strong atheism)