My understanding of a Minister of Music is the person who oversees everything that happens in the Music Ministry. The particular part they perform themselves may vary.
I'm the MoM at my home church. I don't play the organ any more, we have someone else who does that, and he "answers to" me as the MoM. I do the thing that I'm best at, teaching and directing the choir (I also sing backup on the Praise Team). I choose the music and give the musicians guidance about how I want them to sound on it. I give direction (to a small degree) to the sound technician, but not much since I'm still growing in my understanding of the sound equipment.
At the other church where I work, the Music Director (MoM) plays bass in the band. He hired me to direct the choir. He decides what songs the choir will learn, and then it's my job to study those songs and teach the parts. The musicians learn the songs and play them. The boss plays the bass.
So I could imagine a scenario where perhaps the MoM might not do anything themselves, just make decisions and supervise the people who perform the tasks in the music ministry. I don't think that would happen very often, but it would be possible.