Folly, you are describing the difference between weak and strong atheism.
Agnosticism says "we can't/don't know that (well)." It is not exclusive to atheism! Agnostic atheism says "it cannot be proven, and I do not believe." Agnostic theism (very rare) says "it cannot be proven, and yet I believe." People don't like to admit that they are irrational, so this last group is uncommon. People like to say that their religion is objectively true and obvious, and that there are no uncertainties in their religion, so this last group is uncommon.
Atheism is the lack of belief in a god.
Weak atheism says "pragmatically, we might as well act as if no gods exist." Hitchens put it well: "What can be asserted without evidence can also be dismissed without evidence."
Strong atheism says "this god does not exist." When applied to the concept of metaphysical entities as a whole, this cannot be proven. However, it is possible to prove, say, that a god cannot exist that is both the smallest and the largest.
Ignosticism says "what do you mean, 'god'? There's no clear definition of your god." Ignosticism is basically agnosticism+ (not to be confuse with atheism+, which just sucks). As an example, I am agnostic about there being a teacup orbiting Jupiter. I am ignostic about a teacup made out of pure neutrons and yet also blue, orbiting Jupiter but also Proxima Centauri, and nonexistent but also capable of being touched.
Apatheism says "I really don't give a shit." There's really nothing else to say about apatheism.
Antitheism says "religion is bad." It is not necessarily coincident with any other position, nor exclusive to any other position. I can believe in a god and also believe that religion is harmful.