Both of the answers are fine, but I think the words are used a little differently in the past tense. I would say, „I saw that movie already.“ You could possibly say „I watched that movie already“ and everyone would understand you, but for some reason I think most people in the US would say „saw“ in the past tense. I don’t know why. Watch seems to suggest to me the present or future for some reason. See works too for the present, so in that sense you can use either.

For movies and television shows, both "watch" and "see" work. You'll get the point across whichever you choose, but sometimes it sounds better to use one in specific. When going to the movie theater, you should use "see". Example: (Let's go see a movie) = Movie theater (Let's go watch a movie) = Could be anywhere! If adding "-ing", the meaning changes a bit. Example: (I'm watching a movie) = I'm CURRENTLY watching a movie. (I'm seeing a movie) = I'm GOING TO see a movie For bands and performances, use "see" Example: (I am going to see a band tonight) (I'm going to watch a band tonight) = incorrect but gets the point across. Hope this helps a little!

Both could be used in this context.