Whether someone monitors calories probably varies widely. I think most people go through stages, just like with weighing themselves, where they measure their calories and then they give up because it’s really a lot of effort. As a skinny person, I keep my calories in mind but I don’t keep track of them or have a real limit like I did when I was fat.
If you are trying to shed body fat, yes, you might want to switch to leaner meat. Fat is the most calorically dense, having ~9 calories per gram while protein and sugar/carbs are ~3 calories per gram. Protein metabolism is higher priority than fat and more thermogenic, meaning more calories will end up being generated as body heat; incidentally, this heat is mostly created by fat cells which use their glycerol to generate heat since that’s the most proximal source of energy, so you will actually burn more fat if you reduce your fat intake and increase the protein.
I’m just some guy, so read up about “oxidative priority”.