Atheism is holding a definite belief that there is no God.
There are shades of atheism just as there are shades of agnosticism. There’s the de facto atheist who feels that he can’t know for certain but thinks God is very improbable, and lives his life on the assumption that he doesn’t exist. Then there’s the strong atheist who feels that he knows there is no God, as you said (except note I said "knows," not "believes"). Very few atheists are strong atheists. Even Richard Dawkins says he’s not that kind of atheist. I myself am a de facto atheist, except when it comes to the God of the Bible (or other religious gods). Then I’m a strong atheist, and I’m sure Dawkins and Hawking would say the same about themselves.
Either you believe or you don't, there isn't anything in between. Agnosticism just means you accept that there can ultimately be no definitive proof either way.
Like atheists, agnostics come in different shades too. There are those who are very uncertain, but are inclined to believe in God, and those who are uncertain, but are inclined to be skeptical. Those two shades of agnosticism are usually the domain of people who have had some level of religious instruction, indoctrination, whatever, at some point in their lives. They went to church as a child, maybe they still do on special occasions. Religion is a part of their culture or family. Hence, it would be harder for them to totally come to reject it than for someone with doubts, or with an increasingly scientific outlook, who didn’t have that religious background. Such a person would either become an atheist or become a true agnostic who is completely impartial. They have no religious belief. It’s a philosophical issue for them. They (rightly) consider the question answerable and don’t think God’s existence has been proven or disproven. Very few agnostics are of this kind, though the religious ones might claim to be in a debate with atheists just to intellectualize their belief or trace of it.
The reason I’m not of this kind is because I understand that God’s existence or non-existence is in principle unprovable, yet not equally probable. Probablity leans, to my mind, more to his non-existence. (And, of course, the God of the Bible has no probability of existing whatsoever.) Hence, I’m a de facto atheist.