I have no clue if it's what projectivists get at, so you may want to elaborate :P

Projectivism is, I suppose, a part psychological and part meta-ethical theory that suggests people talk about their own desires about how the world should be as if they are objective, mind-independent moral truths. Hence "my morality" -> "morality".

I'm implying that people first noticed what influenced them, and then decided to call parts of it "morality." Thus making it no great mystery that morality influences people. The puddle was shaped to fit the hole, so it has no right to be surprised when it finds itself in a hole that fits it.

That makes sense. But that implies a desires-based theory of moral motivation, which isn't usually considered moral realism.