Earlier quoted context omitted.
I think "They wanted an engineer to build a chatbox that called ChatGPT with company documents as prompt context" fits the term "AI Engineer", personally - see https://www.latent.space/p/ai-engineer which uses it for "applied" AI.
Why would a self-described "AI Engineer" be any more capable of building that sort of functionality over any other backend engineer, especially one who is familiar with agent-assisted development?
It's a speciality, just like being a payments engineer who integrates with systems like Stripe is a speciality.
Being familiar with agent-assisted development helps a little bit because at least you understand prompts, but there's a whole lot more to building software on top of LLMs than that.
Any engineer can get familiar with these things of course, just like any engineer can figure out what it takes to work on payment systems.