The US add engineer to everything to sound most prestigious than they are. If you sell your service as a AI prompt writer, you get paid peanuts. If you sell the same service as AI prompt engineer, the C-Suites cream their pants.
I've found it helpful in learning things about languages I'm unfamiliar with, but it seems like saying "I'm an AI programmer" means "I don't really know what I'm doing in this language, I'm still learning." Which I suppose shows a willingness to learn, but that's about it.