My understanding is that the term's definition, like many words in English/American English, has evolved over time along with how we create music.
The original "music producer" was the guy (it was a sexist era) who ran the radio show on which the artists appeared. He would be presented with an artist, a group of professionals who did the coordination (the director, ad, etc),technical,( the engineers, etc), writers(the announcers would need patter, often the artist may have to plug a product which would have a script), assistants, etc. He would be the guy who would choose those who would do the actual creative work, then critique the result with notes to the director until the performance fit whatever he was paid to make it conform to.
When music recording became separate from the radio companies the recording "industry" started IMO. At that point the "producer" was chosen by the record company, worked for the record company or worked for one of the new fangled "recording studios" that had either been spun off of radio/tv companies or had been supplying content for them. The producer role was similar to its previous iteration, however with some changes. Now he is beholden to a broader range of people, including the artist, increasingly. Now a "vision" for an album and individual performances were coordinated between the artist, their management, the record company, and the producer. The producer did whatever was in their power to achieve that vision to the best of their ability, which varied from one producer to the next. As has been mentioned , some producers have added their musical skills, arrangement ability, engineering skills, and other creative skills. Others just did the critic thing, chose those that would do the arranging, engineering, etc. or just less direct involvement.
Now we have left much of the big studio/record company paradigm behind and multi hat nature of music production has again evolved our definition to include those who create "content" of any kind.
IMHO the definition of producer has just expanded over time to contain all of these possibilities.