I thought we'd get through a whole day without having the atheist/agnostic definition argument. Silly me.
Putting it as a Venn diagram. You draw one circle and label it theism. Then you stop. The negation of an element isn't another circle, it's everything not inside the circle.
This, this, a hundred times this. In exactly the same way, for those people who believe that there is not a god, there would be another circle. Two beliefs, two circles. Everyone who is not in the circle including the theists, they are not theists - atheists. Everyone who is not in the circle including the postive atheists (or strong atheists or antitheists or whatever word you like) is not a positive atheist.
It's about recognizing that "I believe that god(s) exist." and "I believe that god(s) do not exist." are two separate claims.
A comparable situation for the armchair jury - Did OJ do it?
Position 1 - The theist, makes a claim that the event did occur.
"Yes, OJ did it. I vote Guilty."
Position 2 - The atheist, does not accept the claim that the event occurred.
"There was not enough evidence to prove that OJ killed her. I vote Not Guilty."
Position 3 - The positive atheist, makes a claim that the event did not occur.
"No, OJ is innocent, he did not do it. I vote Not Guilty."
In the case of both positions 2 and 3, they do not accept the claim that OJ did it. Position 2 may not have any strong belief one way or the other, but because he has no reason to believe in guilt, he votes Not Guilty. In the case of position 3, they make a further claim that OJ did not do it.
Substitute "God is real" for "OJ did it" and you're done. Positions 1 and 3 make counterclaims, position 2 is the group of people who did not accept position 1's claim. Everyone who holds position 3 also holds position 2.