No, I think he cares, this is not something he wants Thor to find out because he knows damn well that it will destroy the relationship, which he quite likes having now. It's just that he apparently has some lingering effects from the Inversion, where he is actually completely incapable of lying, even if he wanted to.

And let's be honest, it's Loki that did something terrible to Thor more often than not. In this instance, Thor will probably take what he did, murdering his little brother who he thought he had finally redeemed, quite hard. It's probably that which Fury told Thor, and it's just being DEALT with here. I mean, Thor obviously already knows at least the basics if he comes to Loki's place to talk about it, not the other way around.