The etymology is about belief rather than existence.
Theism is
having a belief in the existence of a deity or deities. So strictly, a-theism is
not having a belief in the existence of a deity or deities, which is subtly but significantly different from
having a belief in the
non-existence of a deity or deities (i.e. believing they don't exist).
Current popular usage, and many dictionary definitions, tend to equate atheism with a belief (in the non-existence of a deity or deities). Most atheists who express an opinion seem to prefer the etymological version -
not having a belief - also known as 'implicit', 'negative' or 'weak' atheism, which makes it an umbrella for a number of subsets, including agnosticism (believing we can't know if deities exist), and 'explicit', 'positive', or 'strong' atheism (believing deity or deities don't exist).
An exhaustive (exhausting?) list is given here:
17 Kinds of atheism.
E.T.A. In case you're wondering, I put myself in a subset of category 4 (Difference in the Assessed Rationality of Theism) that isn't on the list, which I call 'introvert atheist', meaning I don't think
I have any justification for belief in a deity or deities, but I'm agnostic (but pessimistic) about other people's justification for such a belief. I guess that means the list isn't exhaustive after all...