Need these two be interpreted in the same way?
(A') I could believe that a unicorn exists on the Mongolian plains.
(B') It is possible that I believe that a unicorn exists on the Mongolian plains.
Now, as a matter of fact, (B') is false given that I believe no unicorns exist, and recognize that this implies that none exist on the Mongolian plains, and don't believe explicitly contradictory things. But (A') is true, because I could believe it, even granted that (in fact) I believe no unicorns exist and recognize that this implies none existing on the Mongolian plains and don't believe explicitly contradictory things. For (B') to be true on the suppositions, there must be a possible world in which the suppositions are true but "I believe that a unicorn exists on the Mongolian plains" is also true. For (A') to be true it simply has to be true, in this actual world, that it would be possible for me to believe that a unicorn exists on the Mongolian plains even though the suppositions are true. Indeed, the fact that we are able to make sense of this 'even though' shows that we have to be able to distinguish (A') and (B') in some cases.
(A') basically says: With regard to me this is not impossible: believing that p. (B') basically says: This is not impossible: I believe that p. Colloquially we can treat them the same, but when we are more precise the two are not really the same, and they don't combine with other propositions in the same way. There is a set of propositions S such that S + (B') leads to a contradiction, but S + (A') does not.
Presumably there is some connection between (A') and (B'). But it isn't, I would suggest, equivalence. If this is true, it would seem to be quite general.