Oi vey! Here we go again.
I know. I always find it curious when people like yourself pull out a brief dictionary definition as if I didn't speak English or as if the purpose of the dictionary was a serious discussion of any topic.
a = without
theos = god
ist = suffix denoting action, condition, use, devotion, or characteristic
But even without knowing the meaning of the word, and granting the simple definitions you have given, a person of moderate intelligence could readily see that "the belief that God does not exist" does not necessarily mean a system of belief, a complete denial of all things religious, certainty that there is no god, a belief that one has "solved the problem of existence", or any of the other nonsense that has been posted here.
If anything, the passage you quoted from Huxley indicates that it's the agnostic who professes a certain belief -- namely, the belief that questions about life and god, about "the problem of existence," are insoluble. If we follow Huxley's definition faithfully, an agnostic is not someone who does not know but someone who believes it is impossible to know. It is the agnostic and not the atheist who has "faith," who professes to know. He knows that "the problem" is insoluble. Atheism makes no positive claims.
Some atheists -- a tiny minority, but some -- do make positive claims, but you can no more generalize from that to say that all atheists make such claims than you can generalize from Christian doctrine to say that all theists believe in the Trinity.
The atheist may or may not hold a particular belief or follow a particular system. The agnostic must, following Huxley, believe in the insolubility of "the problem."
As such, it's not unreasonable to say that an agnostic is a particular kind of atheist, an atheist who professes a particular dogmatic belief that is not shared by all atheists.