Either one is correct. You can also say "I have watched that movie". I remember watching that scene. Yes, I have seen that sceen, and it was a "tear-jerker". It caused me to cry.

scene - part of a film So you can say "I have watched/seen the scene" and it means you have seen the scene, not the whole film. I guess.

"To see a movie" is an infinitive phrase while "I want to see a movie" is a complete sentence or thought. "I have watched the scene" is a different verb tense. The first suggests you have not yet watched a movie, but wish to, whereas, the second is past tense meaning that you have already watched the movie, a specific movie.