Hi, an agnostic here ... just agnostic.

[Jesster]

For any claim: X, you can believe X is true, believe X is false, or have no belief either way. 

Person A: belief X is true, no belief X is false
Person B: no belief X is true, no belief X is false
Person C: no belief X is true, belief X is false

You can do it, if you want, but I think it's nonsensical to pretend B and C are the same, just because they share "no belief X is true", when B also shares "no belief X is false" with A. B is neither, but shares a commonality with both.

Nope. You either believe something or you do not. There are two separate claims that you are confusing and you've created a false dichotomy out of them. There is a claim that there is a god. There is another claim that there is not a god.

A theist will believe that there is a god.

An atheist will not accept that belief.

You can also accept or not accept the belief that there is no god. Either one can be an atheist.

I neither accept the belief that there is a god or the claim that there is not a god.