Don't forget chords are also written. The chord progression goes with the lyrics & melody as written parts.
A "beat maker" is grouped with the musicians as either a performer, writer or both. And if they're involved with the overall sound, they're also producers. A beat maker may also be an arranger.
Producers don't necessarily make tracks. You can produce a song without touching an instrument. Producers of other genres do it all the time. In Hip Hop, the beat maker & producer is often the same person, but sometimes it's not.
These are all separate things that are sometimes done by a single person. This happens mostly in electronic genres where the producer may have made the beat, wrote the chords, arranged elements of the song, etc. But that doesn't make those things production. It makes the producer those things; writer, arranger, etc.