The Greater Fool
Illuminator
You quoted the explanation, not the definition. The explanation does describe the two meanings that people ever quibble about.Isn't one a subset of the other? So if one definition is valid, the other would automatically be included in it?
I do consider the "a"-prefix to mean "not". So an Atheist is not a Theist, i.e. someone who does not believe in a God or Gods, as being a "Theist" means exactly that.
I think the so-called double definition of "Atheist" stems from a misunderstanding of the nature of "belief" as well as from a misunderstanding of the nature of accepting or not accepting claims. If I reject the claim (e.g. in absence of evidence) that it will be raining later today, I do not automatically claim that there will be sunshine. Or even a lack of rain, for what it's worth.
The definition I gave covers the subsets:
Atheist = a person that does not believe god(s) exist.