Sorry, m. If someone said to me "I don't believe that god exists", I would assume that he "believed that god did not exist."
Good point. A lot of people's problem is based specifically on this phrase. In fact, I, and I suspect many other atheists, never actually make this statement - especially without a context clarifying what is meant by it.
If someone asked me whether I believed in god, I might answer "no." But, that isn't me making the statement "I don't believe god exists." If they went on to ask "so you believe that god doesn't exist?" I would answer "no." In either case, I would clarify what my position actually is.
If I were to state my view without responding to a specific prompt about whether I believe in god or not, I would never just make the statement "I don't believe god exists." It doesn't paint the full picture, nor does it provide any actualy useful information about my position. I'm more likely to talk about atheism as a conclusion of being a rationalist than even discuss it terms of "belief."
Anyway, getting repeatedly stuck on the one phrase "I don't believe god exists," ignores the fact that when atheists do use it here, they are specifically telling you what they
do mean by it - there's no ambiguity involved. So, saying that it's an ambiguous phrase is rather disingenuous, although it's a useful tactic for being obfuscatory.