I say lack of belief only because I think it would be wrong to define atheism strictly as a belief. Quite obviously there's no one definition for the word. Atheism branches, for example, into strong atheism and weak atheism, implicit atheism and explicit atheism, as well as gnostic atheism and agnostic atheism, which is the distinction we're discussing now. (I can't be bothered finding a citation that covers all these distinctions, but you can e.g. check the Wikipedia page on atheism and see the sources.)
I believe, as I think most atheists do, that the vast majority of self-proclaimed atheists are in fact agnostic atheists who don't claim to know that there is no god. The common example is Richard Dawkins. Even he, one of the most "militant" atheists in the public eye, self-identifies, on his scale of 1 to 7, where 1 is a strong theist and 7 is a strong (gnostic) atheist, as a number 6.
This is the reason that in an either/or scenario (which word definition needn't be and very often isn't!), I would opt for the "lack of belief" definition.
Stating that something is NOT real is making the claims that both God is not real and I live my life not believing in god. The use of "lack of belief in a God" doesn't deny the potential of a God to be real, however does suggest that you live your life without the belief in God
The word "Atheism" is a shortening of the phrase absence of theism (a for absence and theism as itself). So essentially it is a lack of belief in god(s). You couldn't be and atheist and believe anything to do with a/the god(s) because it is a direct contradiction. Believing is itself a belief about god(s) and the point of the word is that there is no belief about god(s) or related with anything to do with god(s). For instance, we are all born atheists until we are taught to believe or not, if you were religious and are now atheist, you are simply refusing the stories as evidence, not the god(s).
The belief in no god is defined as anti-theism because it displays in the name the action of being against belief in god(s).
People may have their doubts about a God existing. There may have been a time when they strongly believed in God, but something may have happened to change their mind about it. They may have felt let down or that God wasn't really there for them. So, they might have given up their belief and thought it was all a lie.
"What can be asserted without evidence can be dismissed without evidence." - Hitchen's Razor
Because there is no evidence that god(s) exist, I dismiss the theory of gods.
I am also what some may call a Gnostic atheists as I have concluded god(s) do not exist. To me, it is not a "belief" but an accepted fact.
"‘Atheism’ means the negation of theism, the denial of the existence of God." - Stanford Encyclopedia of Philosophy (1)
"Atheism is the view that there is no God." - Internet Encyclopedia of Philosophy (2)
These are peer-reviewed Websites and not really up for debate.
I hope this clears up some confusion.
I find that the definition of atheism as "belief that there is no go" is often used by theist as myself since it "allows" it to be called a religion or at the very least a belief system. Atheism by this definition also cannot occur until you come across the belief of god in general.
I find that the definition of atheism as the "lack of belief in a god" is most often used by atheist. This definition allows them to say "atheism is the natural state". It also allows them to rebuttal theist claiming that atheism is a religion.
Both definitions are correct, which one you use more often depends on your beliefs.