Most people I know who claim to be atheists seem to believe there is not god, despite not being able to prove it. Agnostics seem to be the most honest ones....
That's a very theistic way to frame atheism. Theism requires active belief in prescribed dogma, whereas atheism can include both rejection of said dogma or simply not subscribing to religious ideas.
For example, many people are vegetarian for one subjective reason or another. Just because I'm not vegetarian doesn't mean I believe being vegetarian is wrong or isn't a thing -- it just means I like eating meat and don't have the same reasons as vegetarians to subscribe to vegetarianism. And I don't think about vegetarianism, like how some atheists don't think about god; this stands in contrast with those who subscribe to religion, which requires active belief / prayer / practice / absorption of certain ideals into your identity.
That's interesting. I know plenty of people who label themselves as atheists, but AFAICT not a single one of them take the position of "belief in god not existing".