I share Maverick Philosopher's distaste for the predicate "true for". Phrases like "that's true for you but not for me" are borderline nonsensical. Truth doesn't bend itself to match our beliefs, and it doesn't show different faces to different people. Truth just is. If you want to talk about what different people believe, then damn well use the word "believes", instead of trying to pass it off as a plurality of truths.
Extreme relativism is self-defeating of course. Take the thesis 'All truths are relative' - is this claim absolutely true, or only relatively true?
But not all relativism is nonsense. So long as you have genuine (non-relative) truth at the core, it is possible - perhaps even necessary - to build some (limited) relativism up around this. The key here, though, is not to relativise truth itself, but rather, the framework against which the truth is being judged.
Firstly, note that you can have objective truths about relational properties. It is absolutely true that Jupiter is larger than Pluto. But 'larger than' is a relational property. Moreover, the very notion of 'largeness' (for example) is context-dependent. Is an elephant large? The question is incoherent without a basis for comparison (though in reality we can ask it sensibly, because context should make the implicit basis clear). Elephants are large relative to mice, but not if we're comparing them to whales.
So you can see that we're admitting an element of relativity here. But it's not relativity about truth itself. Instead, it's recognising that the claim being considered is, by itself, incomplete. We need an extra parameter to plug up the gap and clarify exactly what proposition we are refering to. The truth of the claim will vary depending on what parameter is chosen, but this is because the choice of parameter affects the meaning of the sentence. An ambiguous sentence could refer to any number of distinct propositions. But each proposition has a definite, objective truth-value.
Truth is not, ultimately, relative. But it sometimes appears that way because a sentence may be ambiguous or incomplete until 'filled in' by a parameter.
A good example of this is value-judgements. Some people think that value judgements are purely subjective, and simply refer to the individual's own subjective assessments, rather than any objective facts. I am convinced that this is the wrong way to understand it. (I've previously wrestled with some sorts of truths which seem to be inherently subjective, but value-judgements aren't one of them!)
A better way to understand the relativity of value is to say that value-judgements are made relative to some framework of standards. The claim "Beethoven is better than Britney" is, by itself, incomplete. However, once we clarify that the missing parameter is the set of standards shared by classical music lovers, then we have ourselves a proposition which is objectively true. If a teeny-bopper disagrees, claiming that Britney is better, then - according to the above standards - she is wrong. However, if we were to choose an alternative parameter - say, the aesthetic standards of pop music (such as they are), the tables would be turned: the teeny-bopper would speak truth, and the old fogeys falsehoods.
I hope it's clear enough what I'm getting at here. Value judgements, understood as assessing something against some particular standards, are objective. What's relative is the choice of standards. (I made a similar suggestion in the comments thread of a recent FBC post on aesthetic relativism.)
In my old post on Semantic Contextualism, I argued for what appears to be a very broad form of relativism. It is fully consistent with what I've been saying here, however. In effect, I was arguing that there are many truths which are only indirectly related to the objective world (e.g. truth in fiction). When assessing a truth-claim then, we need to consider what world [the parameter] it applies to. Ultimately, however, all truths can be understood as objective descriptions of reality. It's just that they might require some clarifications in order to get to that stage.