We seldom mention the action of starting the movie in the CD player. We just say, "Let's watch a movie".
Even if someone were in the act of putting the CD in the player and we said, "What are you doing", the answer would be, "We're going to watch a movie".
I haven't watched TV for years, but when I did, the act of selecting a movie, as opposed to watching something else, could be called either one. (I didn't have cable until the mid-70's!) Let's turn on the news. Let's turn on a movie. (Even when a sit-com was already playing.) Let's put on a movie. I suppose these days, "Let's put on a movie" means "Let's put one i