The question is usually asked as though it has a yes or no answer. It does not, because “prompt engineer” never named one job. It named a bundle of four quite different activities that happened to be new at the same time, and the bundle came apart.

What the title named

When the term appeared, working with a model meant discovering, by experiment, how to make it do a thing at all. The people doing that were doing four separable activities at once:

Discovery. Finding out what a model could do and what phrasing unlocked it. Genuinely novel work at the time, and genuinely valuable, because nobody knew.

Craft. Writing the actual instructions well — unambiguous, ordered, with examples that carry the format. This is a writing skill more than a technical one and it is still real.