I don't think it's too controversial to expect the language and labels we use to tell us something substantive about what we believe. As it stands, though, the words "atheist" and "agnostic" don't really tell us much at all. Many atheists often argue that everyone who isn't a believer is an atheist, including babies and many who identify as an agnostic. But this muddles and confuses what really should be two separate positions on the existence of God, uncertainty and disbelief. I think we're better off keeping them separate.
Knowledge vs. Belief
The argument goes that atheism and theism describe what you believe, while agnosticism and gnosticism describe what you know. Since most agnostics don't actively believe that God exists, they should be considered atheists -- even if they don't know one way or the other. But even though people treat knowledge and belief like they're very different things, they're really not. People like to construct infographics where belief and knowledge are orthogonal to one another, but knowledge is actually just a specific kind of belief.
This conversation often gets confused by the degree of confidence we have in our beliefs. Knowledge isn't determined by our confidence, though -- knowledge is simply a belief that is both true and justified (Gettier thought-experiments notwithstanding). It just so happens to be that we tend to be confident in beliefs that we think are true and justified. So confidence isn't a criterion for, but rather a consequence of, knowledge. Similarly, many like the distinction between strong and weak atheist, or gnostic and agnostic atheist, because they reflects degrees of confidence. But why should we capture confidence with our labels? We don't do this with any other ideology -- there are no specific labels for a strong Catholic and a less certain Catholic. They're all just Catholics.
So knowledge about God's existence isn't useful in a label at all, because it doesn't tell us anything about your belief -- few people would admit their beliefs aren't justified, and whether or not God exists is completely independent of your beliefs about it. Whether it's knowledge or belief, your attitude towards the existence of God is exactly the same.
Attitudes toward an idea
There are really only three ways to describe an attitude toward an idea -- you can think it's true, you can think it's false or you can have no position at all.
Take the following statement: there is an even number of women named Mary in the state of Connecticut. You can believe that this is true -- maybe you did a census or something. You can also believe that it's false, because maybe the census showed an odd number of Marys. But you can also have no belief about the number of Marys at all -- I'd imagine this is where most of us stand on this issue. Colloquially, you'd just say "I don't know." In other words, you could say you're agnostic about it.
We talk this way all the time, so why make it more complicated for God?
Agnostic-atheist is a clunky, awful and vague label
I've often seen atheists argue that agnostics are really just atheists, based on an awkward definition of atheism as "a lack of belief in a deity." This is sometimes based on something of an etymological fallacy, noting that the prefix "a" signifies negation, therefore "a-theists" are simply those without theism. Agnostics would technically qualify as atheists since they aren't believers. But I suspect few making that argument would also argue that atoms are thus indivisible because the suffix "tom" refers to the word "cut" in Greek. The etymology of a word shouldn't strictly dictate how we ought to define it.
Historically and philosophically, atheism has distinctly been the belief that God doesn't exist (The Stanford Encyclopedia of Philosophy defines atheism as such, which is about as definitive a source as you could want). There have often been those that have argued for atheism to capture everyone except explicit theists, but it seems useless to group people who think there are an odd number of Marys in Connecticut with people who don't believe one way or the other, so why do we lump atheists and agnostics together?
If you tell me you're an agnostic-atheist, I don't learn much about your attitude towards God -- assuming I don't lump you in with the original but outdated definition where agnostics believe that the existence of God is in principle unknowable. I really only learn that you're not a theist -- how can an ostensibly more specific label tell me less about what you believe?
Atheism is better as a label for a specific position
Do you think atheism can be right or wrong? Then you can see where I'm coming from. There's no sense that "lacking a belief in a deity" can ever be true, because it doesn't make a specific claim. But if we stick with atheism as the belief that God does not exist, then it does. It's specific, concrete and, most importantly, potentially true or false. It puts something at stake for atheists. Instead of simply saying "there is no evidence for theism, the burden of proof is on them" we have to say "here's what we believe, and here is our evidence." It's what atheists have been doing for centuries, and it's a cop-out to define away responsibility for our position, just so we can artificially bolster our numbers with vague labels.
That leaves us with three commonsense, simple and (somewhat ironically) more specific words for our religious discussions. Theists believe that God exists, atheists believe that God doesn't exist and agnostics don't believe one way or the other -- there's no reason to make it more complicated than that. Of course, this post isn't meant to be definitive, but I think it captures everything we'd want our labels to capture. I'm open, though, to being convinced otherwise.