Atheist vs. agnostic vs. agnostic atheist

[Tiberius]

I appreciate your clarification, but I think they are answering the same question: Does God exist?

Atheist: No.
Theist: Yes.
Agnostic: I don't know.

You seem to have missed the point of what Simon Moon was saying. The question you just asked was about knowledge (does God exist?). Now, an atheist may answer "no" to this question, and a theist may remark "yes", but the definitions of both atheist and theist do not require them to do so, because atheism is not "knowing there is no God", and theism is not "knowing there is a God". Both atheism and theism are defined in terms of belief.

So whilst your above scenario is perfectly reasonable (you can either know there is a God, know that God does not exist, or not know one way or the other), it's not addressing the belief question:

Do you believe in God?

Atheist: No
Theist: Yes
Agnostic: ...

An agnostic can answer this question either way, because belief does not have to be based on knowledge of the thing you are believing (or not believing) in. I can believe that there it is raining outside even if I have no direct knowledge that it is (but I may have based my belief on other things, like someone I trust telling me it is raining, etc.)

I've already covered how I think the belief question is binary; that you either believe something to be true or you do not. There is no middle ground. One who has no actual formed beliefs one way or the other, by default will not believe even if they aren't aware that they don't believe.

FYI, I answer the two questions this way:

Does God exist?

Tiberius: I don't know.

Do you believe in God?

Tiberius: No.