Caledonian: The statement "x is true" could be properly reworded as "X corresponds with the world." The statement "I believe X" can be properly reworded as "X corresponds with my mental state." Both are descriptive statements, but one is asserting a correspondence between a statement and the world outside your brain, while the other is describing a correspondence between the statement and what is in your brain.
There will be a great degree of overlap between these two correspondence relations. Most of our beliefs, after all, are (probably) true. That being said, the meanings are definitely not the same. Just because it is not sensible for us to say that "x is true" unless we also believe x (because we rarely have reason to assert what we do not believe), does not mean that the concepts of belief and truth are the same thing.
It is meaningful (if unusual) to say: "I believe X, but X is not true." No listener would have difficulty understanding the meaning of that sentence, even if they found it an odd thing to assert. Any highly reductionist account of truth or belief will always have difficulty explaining the content that everyday users of English would draw from that statement. Likewise, no normal user of English would think that "I believed X, but it isn't true," would necessarily mean, "X used to be true, but now it is false," which seems like the only possible reading, on your account.
Given one, we can infer the other either necessarily or by making a minimal assumption of consistency.
No. A belief can be wrong, right? I can believe in the existence of a unicorn even if the world does not actually contain unicorns. Belief does not, therefore, necessarily imply existence. Likewise, something can be true, but not believed by me (e.g., my wife is having an affair, but I do not believe that to be the case). Thus, belief does not necessarily follow from truth.
If all you are saying is that truth conditionally implies belief, and vice versa, I of course agree; I think most of our beliefs do correspond with true facts about the world. Many do not, however, and your theory has a difficult time accomodating that.
Also: what do you mean by a "minimal assumption of consistency?" It is hard for me to understand how this can be of use to you, if it means anything other than, "I assume that beliefs are never wrong." And you can't assume that, because that is what you were trying to show.