There are people who believe God exists (theists) and there are people who believe God doesn't exist. (Strong atheists). And then there are the fence-sitters who believe neither and haven't made up their mind what to believe. (Weak atheists). Therefore not believing that God doesn't exist (fence-sitter) can't be the same as being a theist. Then there would be no fence-sitters.
Your problem is that you can't separate between theism/atheism and theism/weak atheism/strong atheism and keep mixing them up in your arguments.
The fact that you have to have so many modifiers to the term atheism to clarify what you mean shows that it is not an effective and efficient term.
The better already-existing term for those who are "fence-sitters" is agnostic, which should be treated (IMO) as a separate category, rather than shoehorning them in with those who declare that God/gods do not exist (atheists). Those who are not capable of forming an belief should (again, IMO) should be called non-theists, to differentiate them from those who are capable of forming beliefs, but have chosen not to do so (agnostics).
If there are other distinctions, other terms can be identified to clarify.
Rather than insisting upon a dichotomy that clearly causes much apprehension, recognizing the diversity of positions by clearly separating and naming them would seem to be a more rational approach.