I was curious if trivialism could be interpreted as a system that poses significant distinctions of truth values: for example, one that postulates that some propositions can be true and false, but not necessarily all of them: some of them could just plainly be true. I know that such a system can be formally coherent (after all, there is one that is isomorphic to classical logic), but I'm interested if it has been used in that way.

In that case it's not trivialism anymore, but there are nonclassical logics where some (but not all) propositions are true and false; indeed such things are considerably more popular than trivialism (for what I presume to be obvious reasons.) Graham Priest, for instance, is constantly pointing out that if you drop the principle of explosion it's very easy to have the Liar's Paradox be simultaneously true and false without implying that Socrates simultaneously is and isn't mortal.

But this, I get, is not trivialism.

You get correctly, yes.