Information overload. Let me back up.
First things first -- truths (satyas) are objects of cognitions -- which can be either correct (ultimate) or false (relative). Since you are studying Gelug influenced discourse, this may not be immediately evident to you.
I'm sure many things are not evident to me, probably a good thing overall since it means less parikalpita. The best I can do is compare your view with the MS authors, then at least my (mis)interpretations are consistent. Here's what they say about "truth:"
MS pg. 4 wrote:
Buddhist texts sometimes characterize these [two] truths as statements (very roughly, those that are just taken to be true and those that are actually true) and other times as states of affairs or sorts of things (those generally taken to be real and those that are fully real). Because satya means "truth" but also can mean "real" and "what is existent," [we translate satya as "truth" but use "reality" or "existence" when context demands].
So there is a clear differentiation here between truthbearers
(true or false cognitions) and, say, realitybearers
(real or unreal things). I cannot tell if 1) you make this distinction but say "truth" for both flavors of satya
, or 2) your "truth" always means something like "object + a cognition". If yours is the latter, I'll have to work on managing unusual expressions like "true relative truth." (Seem simpler to just say "true cognition" is a truth, "false cognition" is a falsehood...)