I have worked as a script supervisor on TV movies and series for a few years. (That job has nothing to do with writing! It's about keeping continuity. Your screen writer could not be the script supervisor unless he/she was trained as a script supervisor.) Anyway, I hate it when a book about movie-making gets stuff wrong, so I would beta your story as a vigilante against inaccuracy.

I have seen the writer on set when she also has some sort of producer title--Co-Producer, Associate Producer, Creative Producer. I am not sure what factors into giving someone that title, but it could be as simple as a personal connection to an actor that the director wanted for the movie (and probably that actor agreeing to work for a lower rate because of the personal connection.)