@tom_g Let me start by explaining why I take “do you believe in at least one god?” to be a different question from “does at least one god exist?” (note that I have altered the questions from the original forms we were using; I take it that these have the same logical force as the originals). An affirmative answer does little to distinguish them, but a negative answer does much more. To say “no” to the first question is only to say that one lacks a belief. To say “no” to the second question is to state a negative belief (i.e., that no gods exist). They must be different questions, then, if answering each in the negative yields different results. Moreover, each result deserves its own term.
The first question is the belief question, and I do not think it is a mistake to ask it. Nor do I think it is a mistake to find it important. My contention, however, is that it only that it does not distinguish theists from atheists. Your original post presented the belief question as if it had three answers. It seems to me, however, that we now agree that it has only two answers: one either has or lacks a belief in at least one god, thus the answer to the question “do you believe in at least one god?” can only be “yes” or “no.” These two possible answers distinguish two camps: those who are theists, and those who are not theists.
It is true that denying the existence of at least one god, suspending judgment with regard to whether or not at least one god exists, not caring whether or not one god exists, and rejecting the question as lacking a cogent answer all entail not being a theist. There are many ways of not being a theist, however, just as there are many ways of being a theist. My suggestion is that the belief question distinguishes theists from non-theists, but does not yet distinguish what sort of theist (monotheist, polytheist) or non-theist (atheist, agnostic, apatheist, ignostic) one is. Those further distinctions require further questions (“just one, or more than one?” in the first case, “for what reason do you lack a belief in at least one god?” in the latter case).
In short: your “godists” are theists, but your “nogodists” are not atheists—they are non-theists. Atheism, I want to say, is what we get when we say “no” to the second question. To answer the question “does at least one god exist” in the negative is to say that no gods exist. This is a much more robust position than mere lack of belief, but it is not so robust as to rule out the possibility that one could be mistaken. One can assert that no gods exist without appending “and nothing could possibly show otherwise,” thus we need not think that this understanding of atheism involves an assertion of certainty that no gods exist.
Here we might ask several epistemological questions: is the claim that no gods exist merely a belief, or is it something known? if it is merely a belief, is that because it cannot be known that no gods exist? if it can be known, can it be known with certainty? Here, if we were starting over with words, is where I would suggest that talk of “weak” and “strong” atheism would be useful. Not with their past meanings, but to address these subordinate issues. Were I rewriting the lexicon, I would make the following preliminary suggestions:
• Very weak atheism: “I merely believe that no gods exist, and I do not think it can be known that no gods exist.”
• Weak atheism: “it is/may be knowable that no gods exist; but at present, I merely believe it and not claim to know that no gods exist.”
• Strong atheism: “it can be known that no gods exist, and I do know it; but while I claim to have knowledge that no gods exist, I do not claim to have certainty.”
• Very strong atheism: “it can be known with certainty that no gods exist, and I do know it with certainty.”
To sum up, then: we can fulfill your need for words to describe those who believe in at least one god and those who do not with the terms “theist” and “non-theist,” and these are the terms that cover the concept of belief as such (that is, considered as a binary of “has belief or lacks belief”). “Non-theist” has the advantage over “atheist” for this purpose because “non-theist” implies only a lack, which is precisely what we are trying to indicate. It also allows us to keep the word “atheist” in its traditional role while giving us a way of expressing what atheism, agnosticism, apatheism, and ignosticism have in common without conflating any of them with one another.