Good overall write up.
I got a co-production credit unintentionally because I was listening to records and I told someone I was working with that it would be a fire sample. They chopped it up and made a beat out of it, and it later became a song. I didn't even know I could've gotten a credit or anything in that situation, but they let me know that without my idea of using that sample then it wouldn't have happened. I never thought of it that way until I was told.
Producing doesn't equal beatmaking. Often in hip-hop beatmakers end up being sole producers, but you definitely don't have to make beats to be a producer in the official sense of the word.
I've explained it like this to people: If you're in a creative space with anyone that's working on music, and you give any kind of tangible input that either directly or indirectly contributes to the music being made...then you helped produce or write the song, period. Maybe you get a production or writing credit--or maybe not. But you still had a hand in it. This could be humming a bass line, suggesting someone use an 808 and they do it, throw a punchline out there that someone uses in their bars, etc.