Yes, we’re likely talking about different things, and I’m trying to get clearer on your view.
You say you’re “thinking of a possibility of a god that we cannot define.” That strikes me as oxymoronic. The word “god” would be vacuous if it were applied to that which isn’t defined. Luckily for the atheist, the word “god” has a specific meaning, at least for the theist, mysticism being another matter. The founders of the US were deists, not theists, so their deity was indeed less defined than that of the theists (although see Matthew Stewart’s book, Nature’s God; he points out that the founders may have been more like pantheists in Spinoza’s sense). Still, the founders’ God would have been a creator and an intelligent designer, which means their religious views were defined rather than entirely mystical.
Another question I have is about your talk of what’s “unknowable.” There are at least two kinds of knowledge, empirical and analytical. If your point against atheism is that we can’t “know” whether God exists, because we can’t empirically investigate the matter, I’d agree more or less, although I’d point out that empirical knowledge is probabilistic, not absolute. So God’s existence might be improbable, based on the available data.
The other kind of knowledge has to do with our clarity about our concepts. The theistic concept of God is incoherent, as I said. Thus we can know that their God is at least improbable, which again makes for reasonable atheism.
I agree, more or less, that it’s fair “to not make a claim about the unknowable.” Mind you, I’d take this in the direction of cosmicism with H.P. Lovecraft and the like. If human cognition is limited compared to what there likely is in the world, that puts us in a precarious position. The world turns out to be inhuman rather than something made for our benefit. So the alienness or unknowability of fundamental reality would have social implications. We couldn’t know the positive nature of the unknowable, but we’d know the negative implications such as that anthropocentrism is foolish.