Not really because that could suggest that there is God, but you don't believe in Him. I doubt atheists would accept that.
From the theist's own point of view what the atheist accepts or not is irrelevant: the theist, as webs suggested, would consider even the atheist to be "with God" - or at least God to be with them.
And the only difference, from the theist perspective, is the belief.
Existence has to play a role in the definition of atheist, in order to argue against God.
Surely existence plays a role in the definition of atheist because it describes their position: lack of belief in the existence of God.
You make it sound as though atheists were looking for an argument against God and couldn't find one until someone suggested putting the issue of existence within their label.
This seems... an odd view.
The label was coined to describe their position, not formed "in order to argue against God".
It is more plausible to argue against the existence of God.
More plausible for who?
Than what?
It levels the playing field, as the theist will never be able to prove that God exists in the way that atheists define existence. So existence plays a very important role for the atheist.
You seem to see it as an "us against them"?
Your view seems to be that people see theism and think "I need to argue against that, against the God they believe in, but I need to find a reason to do so... Hmmm, let's use the colour of the theist's hair against him... No, that won't work... What about the inability of God to perform parlour tricks on demand? No... That won't work... What about existence? Yes! That's it! I'll incorporate matters of existence into my beliefs and arguments so that I can level the playing field!"
This is absurdist nonsense, I'm sure you'd agree, yet this is what you are suggesting has happened.
Tell you what, why not debate the position that the person holds rather than the label they use (which you might well understand differently to them: why you think they use it, how you think they have changed it to try to get one over on the theist etc).
The atheist's position is what it is, and the label just happens to fit.
It simply means that they lack belief in the existence of God.
Everything else about them is up for grabs.
It just seems natural to believe in God.
,..
Sorry for not giving more detail, but I don't think there is more that I can say that would give you a better sense of why I believe.
It also seems natural to breathe, and takes no intelligence to do so.
We do, however, know why we breathe, and that knowledge takes intelligence to understand.
So why do you believe, Jan?
Is it just a non-intelligent instinct on your part, something that "seems natural"?
Or is there some intelligence behind why you believe, and if so, what is it?