I have always considered myself an engineer because I’m part of a multidisciplinary engineering organization designing a physical product that has embedded software. And “engineer” is the word at the end of my degrees, I guess.

But if somebody called me by any of those terms in the OP I would answer. And if somebody who works on an app or a video game calls themselves an engineer, it wouldn’t raise an eyebrow.

My only conclusion is that we here, who spend our days specifying exactly what we want computers to do, are not so great specifying ourselves exactly.