When "atheist" is defined as "a person who is not a theist", then there is necessarily no overlap between "atheist" and "theist"..
It's unclear, given the references to defining atheism in terms of a lack of belief, if this is indeed the definition of atheism, but perhaps if we start anew with the clearest definitions you've provided, it may help.
Also, it seems reasonable that atheism and theism should not overlap.
However, defining atheism this way means that in order to be an atheist, there must be a way to know one isn't a theist.
If we use only self-definitions, then one can believe in god and be an atheist as long as one believes one isnt a theist. Clearly, in order to define an atheist as not a theist, we need to define a theist. Its pretty uncontentious to say theists believe in god or gods.
The problem is what makes an atheist not a theist when the god a theist believes in, such as human beings or the sun, is something atheists believe in too? That is, if Joe says "I'm a theist" and Jill says "Well I'm an atheist", it follows that there is something that defines Joe as a theist and, lacking that something, Jill is an atheist. Let's say, like a fair number of people I've met, Joe's theism is particular people who have gone through stages of initiation and training without doing anything supernatural, claiming to have any supernatural powers, or in any other way asserting that what they are now makes them different in any scientific sense from the humans they dont considered to be gods. There is in fact nothing about what Joe claims are his reasons to refer to himself as god that Jill doesn't believe in, nor is there a reason to for Jill to disbelieve. The only difference between what the two believe, in order for Jill to be an atheist, is that Joe refers to himself as god.
If Jill is defined as an atheist by not believing what theists do and (apart from a label Joe uses), there is no belief that Joe (a theist) holds about any entity that Jill doesnt hold as well, then how do we determine Jill is an atheist? We could say Joes mistaken, but it has been argued here that we cannot deny Joe's self-designation. So the only thing that can ensure Jill is an atheist is the label, as there is no property Joe has she doesn't believe exists, no entity Joe believes in but Jill does not, nor anything else but a word.
If we are forced to accept the self-designation of those like Joe such that we affirm he is a theist, and are likewise forced to ensure that, in order to be an atheist, Jill must not be a theist, then Jill believes everything about Joe's god that Joe himself does, she just doesn't use that word.
This already makes atheism meaningless, but there are more problems. Because the only motivation we are given so far of Jill's reluctance to use the term "god" to refer to Joe is that she is an atheist. Presumably, though, if an atheist were presented with enough evidence of god they would believe. Because Jill, as an atheist, is defined by not being a theist (and theists are defined by believing in god or gods), shes found a theist whose god she believes in. In fact, as there really are people like Joe, every atheist believes in gods (if, that is, we accept that no matter the nature of the entities a theist refers to as god or gods, that self-designation satisfies the definition). If Jill has no beliefs about gods, and is now presented with one whom her definition of theism demands be considered a god, why doesn't this evidence convince here there are gods? Why, if she accepts the legitimacy of Joe's claim to be a theist and by extension that he must believe in a god, and given that the god he believes in she also believes in, does she determine that she is still an atheist?
Because atheists have conceptions of god and do not wholly define theism by self-description. First, because if theism was simply anybody who described themselves as such then someone who doesn't believe in gods can legitimately call themselves theistic and vice versa. Second, because if we further require that theists believe in at least a god but allow them to determine what they consider god to be, then Joe is a god and Jill believes in him. Were we this limited, Jill would now either have to not believe in Joe's existences (which is ridiculous), or determine that just because someone says they're a theist and/or that some entity is a god, atheists cannot use this criteria and remain atheists.