I'm Japanese(-American) and thus have that perspective on it, more Japan than America.

Being non-religious means you simply aren't concerned. Are there gods? Great. Are there no gods? Awesome. Jesus is the one God? Okay. Zeus leads his pantheon of gods? Nice.

Atheism meanwhile is a deliberate belief in no religion. It's different from simply not caring about religion at all, because you do care about religion insofar as to not believe in it.

That sounds more like implicit atheism, which is a distinction within atheism, but still commonly defined as atheism: https://en.wikipedia.org/wiki/Implicit_and_explicit_atheism

Implicit atheism is still concerned with not believing in religion.

The way I see atheism is that a belief in no religion is, by its nature, a religion. Joe is a christian and believes in Jesus, Bob is an atheist and believes no god; both are merely two sides of the same coin. You can't call yourself non-religious if you believe in a religion, whatever the specific form.

Being non-religious means you don't care; it's not that you don't believe, you simply could not care less one way or another. Wikipedia appears to call it apatheism[1] and more broadly irreligion[2], but semantics aren't the focal issue here.

[1]: https://en.wikipedia.org/wiki/Apatheism

[2]: https://en.wikipedia.org/wiki/Irreligion

Uh no, not in 'Europe'. Where I am in Europe, 'atheism' is being convinced there is no god. Agnosticism is not knowing whether there is a god at all. Not that you're religious but not to a particular god or doctrine like you claim - I've never heard of that concept before, how can you be religious but not knowing what it is you believe in? I'm not much in the know on the exact nomenclature, but your definition of 'agnosticism' is not in any way supported by the way Wikipedia describes it, and the way I've always understood agnosticism is what is called 'apathetic agnosticism' on Wikipedia - and that is, from my perspective, the predominant understanding of it in 'Europe'.

I'm from America and have heard both these concepts, just to add a data point

You're confusing agnostic with what is usually termed spiritual in the UK. People who are spiritual sort of "pick and choose" whatever they fancy, or just feel there's some greater being but don't think any religion gets it right. Sometimes it's monotheistic, sometimes polytheistic.

Agnostic means you believe it's impossible to know whether god exists or not.

I do agree with what you're trying to describe, that many people in some European countries just don't care one iota about religion. I don't think Americans can really understand that without living in a secular society. It's such a non-thing in our lives that, non-religious, agnosticism and atheism, etc. all tend to get mixed into one.

Society has become so secular that those technical definitions have become essentially meaningless. God doesn't exist/I've given it no thought/I don't know if a God exists/I don't acknowledge the existence of supernatural entities all are essentially the same position because it takes up so little of our time or brain power and has so little consequences on our lives.

I stress "some" countries as some European countries, or just small parts of those countries, are still fairly religious.