Why? Agnostics don't believe in gods. Atheists of all shades don't believe in gods. Why not just use the terms "weak atheism" or "strong atheism"? Those are clear.
No, agnostics don't think they can prove or disprove the existence of gods, it has nothing to say on whether they believe or not.
"Agnostic" is used in two* different ways as well. There are those for whom agnositicism means they think it is not possible to prove or disprove the existence of any gods and there are those who just say, "Dunno." For the latter, they may think that it is possible to discover whether any gods exist, but they've not been able to do it.
* Your definition is slightly different again.
Atheists don't believe. If they are gnostic atheists, they actively believe gods don't exist - they think that's demonstrable. If they are agnostic atheists they don't believe, but don't think there's sufficient evidence to conclusively demonstrate that, they simply rest on the provision that the case for gods has not been proven.
O.
OK with that, though I prefer the use of "weak" and "strong" due to possible confusion over the term "gnostic/Gnostic" and I like the idea of calling someone a "weak atheist"