Yes, I understand the arguments around theological noncognitivism, but remember the word god has to have subjective meaning to the individual, not an objectively complete definition that meets your exacting standards to cover every single god concept in the history of the world.
It isn't a subjective/objective problem. Even within the subjective understanding of a single person, there would be contradictions in a definition of "god". This is why, for instance, we can recognize Christians and Muslims as monotheists while also recognizing Pagans as polytheists despite the Christians and Muslims believing in angels that are very similar to Pagan gods.
So I ask you again: define "god". It can be your own subjective definition. You claim to have one, so this should be easy.
You make this same mistake repeatedly when you keep claiming that an atheist (as per the traditional definition) has to be able to understand and dismiss every single god concept ever to be an atheist. They only have to dismiss what they are aware of and personally consider to be god.
But even babies do that much. We're you one of the ones who argue that babies can't be atheists?
So by your approach, atheists almost continuously stop being atheists when they become aware of a new god-concept and stay "not atheists" until they learn enough about the concept to dismiss it?
"I was an atheist, but then I met someone with a new kind of theism this morning that I hadn't considered. I won't be an atheist again until at least Thursday - I'm swamped with work, so that's my next chance to research what that guy was talking about."
- does this ever happen?
It is about their subjective interpretation, not your opinion on whether or not it is objectively justified. Belief, not 'fact'.
The subjective interpretation of anyone who considers conflicting theisms to all be theisms is a noncognitivist interpretation of "god".
If you agree that Christians are monotheists and Pagans are polytheistic, and that pantheists are theists while atheists who believe the universe exists are not theists, then you have a noncognitive definition, too.
But if you disagree, feel free to give your definition to demonstrate that it really is coherent.