Actually, the way I’ve heard atheists and agnostics put it (and it’s only fair to let them define their own positions) is that there are actually three positions:
Agnosticism: God may exist, or he may not exist, but we cannot possibly know.
Weak atheism: I lack belief in the existence of God, but I make no claims about his existence.
Strong atheism: I believe that God does not exist.
A strong Athiest doesn’t “believe” God doesn’t exist. They may use those words, but it’s usually because some bright spark will alway’s say to them, that they can’t ‘prove’ it therefore they don’t “know”. So they say they don’t believe.
But it’s not really an accurate statement. As a comparison(commonly used):
Take Santa. I do not “BELIEVE” that Santa does not exist. It isn’t about belief. Simply said, Santa does not exist.
In the same way God does not exist.
It’s the same thing for an athiest. They don’t “believe” in anything. They don’t 'BELIEVE" that God doesn’t exist, that would imply there’s a choice being made between two alternatives. There’s no Philosophy behind it, and it’s not really a choice.
I didn’t make a choice to believe in Santa or not. He just isnt’ real and I accept it. I lack a belief in any Santa, Fairy, gnome(except maybe in computer games) etc etc.
Another way to think of it, is that for the athiest, there aren’t really two options that believers claim. IE God or No God.
There is only one option.
Another way to think about it could be, there are as many options as available to the human imagination.
There is only one option that is correct, and that is their reality.
Having spent my entire life as an athiest, I understand the mentality very well, though most athiests, since they lack any belief in a God, don’t spend too much time worrying about it nor describing it unless they have a specific interest in history and religion.
For the record, I’m now agnostic and it is a VERY different place to be. Some might call me a weak Athiest, but I’m not.
Rather than worry about definitions however, since we alway’s debate that stuff anyway, I think the most important thing to realize is that athiesm itself, has no “set of rules” or particular Philosophy.
To them, there is no God. That is ALL that it means, nothing more, nothing less.