To see would be translated as thấy and the subtle difference, I suppose, would be that "to see" simply means "to have vision of" while "to watch" is more intentionally/purposefully seeing someone.
Then why does drop down hint offer "see" as top choice? At this point in my experience am I to assume drop down hints are not reliable for correct answers? Then why have them at all?
That's because xem/coi can technically be translated as "see" in some situations. You can see a movie and you can watch a movie. In Vietnamese you cannot "nhìn/thấy phim" as that would mean something different. You "xem/coi" phim = watch a movie. "Tôi nhìn phim" is strange and "tôi thấy phim" just means you either saw it or glanced and had vision of it but didn't sit down and watch it. It can also mean "I find/found the movie [insert feelings about movie]". It's confusing, I know.
Tôi nhìn phim = nobody would say this.
Tôi thấy phim = can mean either "I see/saw the movie" or "I find/found the movie such and such".
e.g. Tôi thấy phim rất xúc động = I find/found the movie very moving/emotional.
Tôi xem/coi phim = I watch/am watching the movie.
NOTE: Xem is more universal while coi is more-so used by Southern Vietnamese.
Thank you for taking the time to provide a clear and informative answer. That helps me a lot. BTW, I took the liberty to nominate you for a moderator.