I agree that the terms fact and truth are used differently, although there's a lot of overlap. Facts are normally fairly objective, requiring little interpretation. Facts are also truths. But some truths are broader than a fact. They can be broader perspectives on life (or science, or other things).
However I resist the concept of things being true for one person and false for another. Obviously people are different, and approaches may work for one and not for another. So there can be things like "this approach is better for me" which is true for some people and not for others. But that statement has a built-in personal reference "for me." So we're not really dealing with relativity of truth. There may well be things we don't fully understand, so we'll have to live with differences of opinion, or with multiple descriptions, each of which capture some of the truth, but in the end I still believe that propositions are either true or false. The type of truth may, of course, vary quite a lot, from "this object is red" to "God sent his son to redeem us."