No, programmers are not engineers, generally, even if it became trendy around the late-90s dotcom boom to give every programming and programming-adjacent job in private industry (government didn't really join this trend) an “engineer” title.
I think an appropriate name might convey a mix of the concepts: problem solving, design, creative, abstract, structure.
If you add "rigor" to this, you have a good general, multidisciplinary, description of an "engineer". I think this slight, but important, difference is why people want to use "engineer" for non rigorous software.
In reality, you probably have some gradient between "software designer" (the ever fluid art and design of software) and "software engineer" (proven critical systems).
Hmm, something like (now hear me out), I dunno, “developer?”