Americans don't want an atheist president

Again, not quite, the "non" in this sense does not mean the person has an opinion on the subject, it simply means they are 100% neutral on it; either they are able to completely block out any and all effects of religion, or it has never even come up in their life so they don't even see it as a subject of debate, it literally does not exist to them.

But, at the point where we're talking about it as a concept the person claiming to be an atheist has obviously encountered the concept of belief in god. The completely untouched "no idea of god" premise is the only other one left by the definition and its not relevant if we're describing atheists in society. Someone who is amoral is not "neutral" they lack morality. To believe in amoralism is to actively deny a role for morality.

Also, you're totally a deist.

Agnostic means just that you're not entirely sure - it's perfectly fine to be an agnostic atheists which believe that there is no God, but don't know if that is true.

That's really stretching things. You don't call a Christian who is having doubts an "agnostic Christian," even though it's obviously possible to have doubts in any system of belief. Furthermore, making the active effort to believe in the absence of higher powers while believing theres no grounding in fact for such an assertion is a little batty. If you're nowhere near sure, why would you tell someone you make an effort to believe the exact opposite of them? pomegranates and giggles? I'm not saying people don't do that, just that when we're talking about what these terms mean to people, and how people should be describing themselves, it's needless complication.

And Christianity "refutes" any other religion too: I don't see Christians saying that Greek mythology or Norse Gods exist.

Christianity disagrees with some tenets of those religions. In the case of polytheistic religions, Christianity says there is one true god. Leaves the door open for false gods, lesser (but still higher than humanity) beings, and everything else that could be "misinterpreted" by other groups throughout the ages. Heck, it's how Christianity spread in large part. "What's that about winter tree festival? Oh, you mean Christmas!" Exploit compatibilities and call other belief structures right to the extent they are similar. The only faith that disagrees philosophically on the most basic point is atheism. Every other faith I can think of has some kind of additional categorical or metaphysical similarity to others.

To quote BSG: "What is the most basic article of faith? This is not all that we are"

The thing to remember here is that the definition of atheism is the belief of lack in a higher power. That doesn't mean they deny that Gods exists, nor does it mean that they hate religion.

It means exactly and specifically that they deny that God exists. If you don't deny(/believe in the absence of) the existence of gods you are not an atheist. Anything else is torturing or outright contradicting the meaning of the word. I'll point you in the direction of bs concepts like "weak" and "strong" atheism, where the belligerent ones who hate religion tried to change the meaning to forcibly absorb agnosticism into their ranks so they could say silly things like a-the-ists (no-god-ists) don't necessarily believe there is no god. These concepts come to us from logic and philosophy so we'd actually be able to describe things accurately and succinctly. To use them to frame an argument dishonestly pisses me off to no end.

A minority of atheists actually bother to discuss religion with other people like religion was a plague - and it sadly seems like you've only met those atheists.

If you believe faithfully that premise A is true, and a large part of the world considers itself part of groups that believe premise A's exact opposite premise Z is true, you'd probably also lament the foolishness etc.

It's understandable, just not as philosophically defensible as they seem to think it is.