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1) While this is probably true for most if not all atheists, it is not the essential definition of atheist. It is a belief.
It is not intended as a definition, since it is incomplete, as can be seen. THis is an enumeration of cases to more clearly explain the point.
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2) This is closer to the essential definition of atheist, but I think the "or do not exist" is superfluous.
In fact the "or do not exist" is essential, because it is a part of the case the point is meant to cover.
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3) Agree. Now add a "not" between the "does" and the "believe" and you have the essential definition of atheist. A non-belief.
Which is clearly illustrated if you take my points as intended and not as definitions.
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Numbers 1, 2 and 3 added by me. I absolutely agree with 1 and 2. They are the essential definitions of theist and atheist.
3 is fuzzy, and I offhand don't know what to say.
Taken in context it seems clear to me that I was using the common definition of agnostics as those who believe neither that a God does exist or does not exist. Using Huxley's definition it is possible to be an agnostic theist (and such people do exist) but it should be clear that I did not intend that usage.
I am not sure that even RAZD really means the usage that you ascribe to him - I think rather that his confusion is based on his inability to understand your definition of "atheist" (which is, I grant, not the common definition but is certainly a valid definition that is used quite widely). Certainly so obvious an error - in the face of many explanations is bound to lead to problems.