No, you've got your definitions a bit wrong. Yes, a theist BELIEVES in a god and an atheist DOES NOT BELIEVE in a god, however, agnosticism and gnosticism has nothing to do with belief, it has to do with KNOWLEDGE. An agnostic either claims that a god is unknowable or that they do not know whether a god exists. There's no belief involved with agnosticism.
And I'd go a bit further and say that most agnostics are in fact atheists. Because when you ask if they believe that a god exists, the answer invariably will be, no, I'm agnostic. They do not believe that a god exists, and thats all that it takes to be an atheist.
I think this double definition: atheist/agnostic or Christian/agnostic etc. should be scrapped. I don't want to hear anyone calling themselves an agnostic unless they really aren't sure what to believe. Because that's what we are essentially talking about here -- beliefs, not knowledge! If someone claims to know, they're just a babbling idiot who should be ignored. Nobody knows for sure whether or not there is an intelligent force that is either part of the universe or acting upon the universe, case closed!
So all this claptrap from Dawkins and his acolytes that goes something like -- my belief claim is atheism, but my knowledge claim is agnosticism -- does is create confusion, and possibly for the purpose of denying agnostics their own separate space, and lumping them in with some broader atheist movement.
Back over a hundred years ago when Julian Huxley wrote an essay where he created the term "agnosticism" to describe his own worldview, there were atheists in academic circles, so why did he go through the trouble of creating a new definition for his own views instead of just accepting the already existing atheist label?
As for whether atheists are closed-minded, that's an impossible question to answer without knowing all of the baggage that comes along with their atheist perspective. As gets pointed out through endless repetition, atheism just defines non-belief, not a belief position; but nature abhors a vacuum, so if someone is an atheist, they still have beliefs that they use to create their worldview. Many atheists are secular humanists or naturalists or existential nihilists etc.
What I've noticed in recent years, with the advent of the internet and online virtual communities, is that there is a loosely organized activist atheist movement which has added extra philosophical baggage that doesn't directly come from being an atheist. For example, the belief that atheists should or need to unite together in some common movement to promote rationalism and work towards ending or at least confronting all forms of religion -- both moderate and fundamentalist, as an enemy force. And that's probably why the vast majority of atheists don't join atheist groups.