>Rather, an atheist believes in the non-existence of gods. We don't need a word like non-theist because we already have a word for it.No. That would be a gnostic atheist. One who claims to know and doesn't believe.
Nearly every single atheist I know, and it is generally a safe assumption to make [0], is an agnostic atheist. One who doesn't believe but claims to not know for sure.
If you ask me if gnomes exist, I will tell you I am a gnostic atheist. I am certain in my knowledge that gnomes do not exist, and thus I have no reason to believe in them. If you ask me if god(s) exists, I will tell you I am an agnostic atheist. I am uncertain in my knowledge that a god does or does not exist, but I have no reason to believe that one does.
And yes - there can be agnostic theists too! Although they are the minority as most theists claim to "know" and that knowledge is why they believe. So theists are assumed to be gnostic theists unless they state otherwise. My grandmother is the only person I've ever known to claim to be an agnostic theist. She claims to not know but finds comfort in believing. She doesn't follow any major world religion and her god is not a god of any scripture.
[0] It's the vast majority and so "agnostic" is usually superfluous and unnecessary in conversation. I only ever see it brought up when a theist tries to claim the Atheist is really just a Theist who "believes in some opposite thing". Then the difference has to be pointed out like I've done in this post.