In English, negation of a modal statement naturally rises to the most controlling interpretation of the broadest phrase into which it might fit: "A does not believe P", without any quirky inflection, expresses "A believes P to be false" -- A believes (not P). Likewise "A does not want P" means A is actively opposed to having P, not that he lacks the wish to have P, but might not be opposed. On the other hand "A does not need P" really means that A is independent of P, rather than that A is dependent upon the absence of P. Whatever form gives the agent the most power is meant by default. So the vocabulary, and not the form, determines the interpretation.
(There are obvious exceptions, but the last statement remains true.
Most ludicrously "You may not do that" forbids you, whereas "I may not do that" indicates my own freedom.)
But that grammatical fact is arbitrary, and the shifting about of the negation makes it hard to proceed in English. So to the degree that is part of the question, it is about English grammar/usage and not about logic.
If we step outside of English it is clearly possible to disbelieve P, or to fail to be convinced of P. In a sort of pseudomathematical form
not(believes(I, P)) != believes(I, not P)
In the broader symbology of logic, at this point one often whips out the 'box' which indicates some special interpretation of 'necessarily', in this case 'is believed'. ('Necessarily' because it is what seems necessary for this to be true in the mental world of the person in question.)
So there is a distinction between
A: ~ p
A:  ~p
The former meaning that from A's point of view p is not proven and the latter meaning that from A's point of view p has been proven false
For every 'box'  there is a 'diamond' <>, indicating some variant of 'possibly', in this case 'would consider'
So the corresponding phrases
A: ~<> p
A: <> ~p
Indicate respectively that A is not open to considering P, (which means he believes ~P) and that A is open to considering ~P.
There is an entire formalism for clarifying such things, the doxastic variant of Modal Logic, which you can look up anywhere.