For those who came in late, this is where I've laid out the definitions of all these terms.
It is indeed possible to be an agnostic strong atheist, Jessica. That means you allow for the possibility that there's a god, but you positively believe there are none. A similar position is actually the exact opposite one. The world is full of agnostic strong theists: those who do not claim certainty that there's a god, usually because of lack of evidence, but believe in one anyway. The very concept of faith encourages this position.
My question to you would be the same as to any theist: "Why do you believe this, when you know it might not be true?" In the absence of hard evidence, what is your alternative basis for believing that there aren't any gods? Consider if you like, and get back to me.
You don't have to burden anyone else with these semantics. Simply calling yourself an atheist expresses most aspects of your position. If people ask you how you know there's no god, you can honestly say that you don't, and that you don't have to. If they accuse you of having as much faith as a theist, they might actually be right (since you're a strong atheist), but this doesn't make you any worse than them.
Atheism is a-theism. Theism is a belief in a god or gods. Atheism is therefore no belief in a god or gods, not a belief in no gods. This means that many people who call themselves agnostics are actually atheists as well. The definitions are not mutually exclusive.
I get your point, and in fact I'm happy to call myself an agnostic atheist. I am an agnostic because I don't know whether there are any gods, and I suspect that I can never know. But I insist to you that I'm also an atheist, because I conclude from this lack of input that there really aren't any. It's a conclusion rather than a belief, and I'd change it if some evidence for gods turned up, but it's my considered opinion that if there were a god, there would be more evidence.
It's like being confident that you won't be struck by lightning tomorrow. You can't say it'll never happen, but it's so unlikely that you live as if it won't happen. In all probability, it never will. If three of your neighbours were struck by lightning in the same week, however, you might reconsider that position.