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You're correct to note that 'it is not the case that X believes Y' and 'it is the case that X believes not-Y' aren't equivalent. Let's be slightly systematic about this. Consider the following four propositions:

Prop(1): 'it is the case that X believes that God exists'
Prop(2): 'it is not the case that X believes that God exists'
Prop(3): 'it is the case that X believes that God does not exist'
Prop(4): 'it is not the case that X believes that God does not exist'

Note, first, that if X is completely unacquainted with the concept of 'God', then Prop(2) and Prop(4) are both (vacuously) true, but Prop(1) and Prop(3) are both false. So, in particular, Prop(4) might well be true while Prop(1) is false, that is, it could be true to say that X does not believe that God does not exist, while at the same time it could be false to say that X believes that God exists (because, by hypothesis, X lacks acquaintance with the notion of 'God' entirely). Clearly, not believing that God does not exist is not in general equivalent to believing that God exists.

But it would appear that the following statements are true:

if Prop(1) is true, then X is a theist;
if Prop(3) is true, then X is an atheist;
if Prop(2) and Prop(4) are both true, and if X is acquainted with and understands the concept of 'God', then X is an agnostic.

Finally, it seems to me that the distinction between a 'positive atheist' and a 'negative atheist' may be captured in the following way:

Suppose Prop(3) is true of X (that is, suppose that it is the case that X believes that God does not exist). Then:
(1) X is a 'positive atheist' if X believes that it's not possible that s(he) could be mistaken in believing that God does not exist, and therefore all attempts to argue otherwise are wrong-headed, and all attempts to educate individuals and structure society in a way that supposes that God exists are mis-guided and ought to be resisted in every way.
(2) X is a 'negative atheist' if X believes that it's possible that (s)he could be mistaken in believing that God does not exist, and therefore tolerance is the order of the day with respect to those who believe that God exists, and, as long as religious beliefs don't have pernicious and damaging effects on members of society or the society as a whole, the holders of such beliefs should be completely free to put them into practice in the manner they freely choose.

There's more analysis to be done, but I'll leave the rest for the entertainment of readers. But be careful: verbs of propositional attitude (such as 'believes') are a bit tricky to deal with.