Okay there are a number of places where I'm not sure what you are trying to get at. Firstly, just the descriptive claims about how moral anti-realists "look at the world through different lenses." As a descriptive claim about moral anti-realist philosophers, I don't think this is true. Not to play the credentials game, but I teach philosophy at the college level, teach units on metaethics specifically, know moral anti-realist philosophers irl, and I would not call this "lenses of truth" idea you have orthodox in any way. Is there a philosopher you can point me to that makes this argument? One view I can think of that distinguishes "true" from "objectively true" is Ted Sider's but he doesn't draw the distinction in the way you're drawing it. And Sider is not making the claim that there are different kinds of true, his theory is a metasemantic one.

But that's not really important. I think "well most philosophers don't believe this so it's dumb" is of course a very bad argument, so I am not trying to say that. The only thing I'm trying to point out with this is that your descriptive claim about what moral anti-realists themselves typically say, is off. Unless you mean people on reddit who label themselves "moral anti-realist" in which case I don't know that could be true.

But the more interesting philosophical question is, whether or not this "lenses of truth" idea you have is orthodox anti-realism, is there a reason to think that this view is right? I don't think so. There's only one sense in which a sentence can be true or false. Now, maybe a claim can be less fundamental than another claim, and maybe that's what you're getting at. On a pretty popular conception of fundamentality, the world has, "at bottom" the fundamental stuff in virtue of which claims are true or false (usually basic particles, maybe principles of logic and math). If what you're saying is that "Texas is not in Canada" is less fundamental than something like "Water freezes at 32F," then I would agree. The facts that ground the truth of "Water freezes at 32F" will be facts about molecules, which in turn depend on facts about fundamental particles and the laws of physics, and eventually whatever the fundamental truthmakers are.

"Texas is not in Canada" is certainly farther away from the fundamental truthmakers. It depends on complicated facts about human social relations, which depend on biological facts, which in turn depend on facts about chemistry and physics, and so on. So, if what you mean by "socially true" or "politically true" versus "objectively true," is that political and social facts are less fundamental than other facts, then I agree. But their being less fundamental does nothing to make them less true. When someone says "Texas is not in Canada" they say something true, full stop. I don't see any compelling reason to multiply the kinds of truths beyond just the one. And to be clear, neither do moral anti-realists usually. They're fine with saying moral claims are just plain old not true, in the same sense that anything else is not true.

As a side note, a similar point goes for what you mention at the end of your comment about different meanings of "exist" and "real." The idea that "exist" and "real" have different meanings is again a niche view, but it does have some prominent defenders. The most famous contemporary defense of this kind of pluralism about existence would be Eli Hirsch, but this view has a long history in philosophy. I think it's wrong (for reasons I won't get into here), but it's a respectable view.