Donald Knuth's great work is called "The Art of Computer Programming."
There's really no answer to this question, but it may help to consider the motivations and interests of the people who use the names. There is a cluster of meanings that hover around the word "engineering" and around the word "art." Programmers fall somewhere in between the two. The search for absolutes is meaningless.
If you consider a symphony orchestra, it performs an economic job. It has to deliver cost-effective music on schedule. Performances contain defects; people need to decide on the acceptable level of defects. The performer are highly skilled operators of machinery like cellos and celestes. There is a management hierarchy; individual contributors, middle management (the "first seats,") upper management (concertmaster, conductor) etc. etc.
Yet few would call musicians "music engineers."
At the other extreme, the person who decides what heating and cooling systems need to go in a new building, and how the pipes and ducts should be sized and routed, exercises a great degree of creativity, but few would call them "HVAC artists."
What can be said is that management wishes that programming were more predictable, more standardized, and less dependent on individual heroics by non-interchangeable, talented individuals. Management is apt to fall for any smooth talker who claims to be able to organize the work of programming to be less like art and more like engineering.
Wishing, however, will not make it so, and the CMM Level 5 firms will continue to produce both good and bad work--healthcare.gov was the product of a CMM Level 5 organization. And meanwhile, both bad and good work will continue to be done by "undocumented, chaotic, ad hoc, reactive" manner by small teams of good people who give a damn.
I always preferred to be called a "programmer" because I have always felt that "engineer" sent a signal that I was working for someone who didn't really understand the nature of my work. I think a violinist would prefer to be called a "violinist" than a "chordophonic engineer."