You're getting to it. There's a distinction between "atheism" as an impartial descriptor of a person's orientation within the greater continuum of belief and a public or personal identification as an atheist. You can point to a person's beliefs and say "he believes there is no god, therefore his beliefs are atheistic". A person can also point to to himself or herself, either publicly or privately, and say "I am an atheist", claiming the word as a marker of identity. This is a complexity that arises when discussing people who espouse any ideology - on the one hand, there is a bald statement of category ("that person is a Republican, because he agrees with Republican beliefs"), on the other, there is an identity which the person allies himself or herself with ("I am a Republican, and I attend party meetings and enjoy talking to like-minded people").
So, in my case, I don't believe in a god or gods or angels or demons or anything like that. I have a naturalistic worldview. I identify as a "bright", because I like the idea of the Brights Project, which seeks to escape the strong negative connotation associated with the word "atheist" and create a broader umbrella term that encompasses agnostics with a naturalistic worldview. Basically, it seeks to do the same thing to unify the non-believing community that "queer" seeks to do in unifying the LGBTQ community. However, there is an "atheist community" of people who embrace the term atheist and use it to identify themselves.
So, you're absolutely right that, by definition, someone capable of holding belief in a god has to sit somewhere as either a theist, atheist, agnostic, or apatheist, but, as zeebra pointed out, people's identities do not fit nicely into these boxes. A strong atheist is what you would call an atheist (positively believes there is no god), while a "weak" atheist is really an agnostic (does not believe in god, but does not believe this with 100% certainty). Even Richard Dawkins admits that strong atheism is indefensible and calls himself a weak atheist (a technical agnostic), yet he still identifies as an atheist. See how confusing it gets?
This is why I objected to just pointing to the dictionary and saying "Atheists positively believe there is no god". I
do not believe in a god, like most atheists/brights, and there is a real distinction that the dictionary definition is not helpful in elucidating.
If you want further reading on this, Richard Dawkins covers it quite eloquently in The God Delusion.
Here's more on the brights, too.