Ultimately, the greater community will define for themselves what constitutes a “musician” or “producer”; as it pertains to their personal goals and ambitions, the company they wish to keep, and whatever audience or clique they choose to pander to.
As such, I’m not going to argue or disagree with what anyone else has said above.
What I can add to the conversation, however, is a reality-check of sorts.
The bottom line is, if you aspire to be a paid professional, within the context of the music business, you will be required to demonstrate a level of proficiency that many music hobbyists might find intimidating or discouraging.
A real music “producer”, not unlike a composer or conductor, is typically the most proficient musician and engineer in the room. That’s why they get the job.
And no “musician” gets to be a session player or hired gun without having mastered an instrument; and that includes having a thorough academic understanding of music theory. To which end, real-world credentials are often required.
But that should not deter anyone from making music and finding joy in it; regardless of their approach, skillset, or level of proficiency.
Cheers!