This is more a language problem than anything else. Yoy’re using the word “truth” to describe honesty, as in “to tell the truth”. This is troublesome because the word “truth” is also often used to describe what you are calling “fact”.
When we say “tell the truth” we mean “do not intentionally decieve”, but this allows for one being wrong, so it’s not indicating reality necessarily, just one’s honest conception of it.
You have made the distinction between truth and fact to resolve this, which would be perfect if not for the fact that it obfuscates the reality due to colloquial use of language. It’s like using the word “God” or “Love”... they are terms already confused and varied in people’s minds. When you say “truth is subjective” many will take this to mean that they are not strictly obliged by reality, that there’s room for interpretation in all matters - a very convenient (and thus popular) quasi-philosophical position - though this is not your intent.
This becomes dangerous as it regards morality (a subject already beset by confusion). If I say “all presidents are immoral”, the relativist says, *”Well, that’s your truth, not mine. I think some are good.” But no, we can’t agree to disagree; we don’t get to have different truths. One of us is wrong, and so our opinion has nothing to do with truth (i.e. fact).
You solved it, but in a perilous way because your terminology is the same as the relativist, though with a different intent. Since they are less philosophically advanced to begin with (as per their belief that “objective” reality doesn’t exist), we can’t expect them to make this subtle distinction. They will hear your words and feel bolstered, seeing you as an ally, and the last thing these people need is encouragement.
Prudence dictates that we insist on the word “truth” being used to describe “that which actually exists and is not subject to interpretation” so as to confront solipsism directly, where it resides, which is in this word “truth”.