If there is a moving image (e.g. you are in the cinema in front of the screen) you would say "watch". If you are just thinking about the film as a work of art, you would say "see", as in your example. If you are referring to a static image like a photograph or a picture you would say "look at".

'I am watching a film...' is more natural/correct, but natives may use 'see' also in the context of films. This may almost be idiomatic because in the past people would only watch films at a movie theater or at some kind of event. An old phrase is 'to see a picture,' where 'picture' is synonymous with film/movie in old- timey lingo. Generally, sticking to 'watch' will be safest, however. The following are valid expressions: 'We're going to see a film.' 'We're going to watch a film.' 'I can't meet you at 5 pm because I'll be seeing a film.' 'I saw that film.' Fun but grammatically incorrect regional expression: 'I seen that film.'

Both can be used. "Watch" is more specific. The relation between "see" and "watch" is like the relation between "having dinner" and "eating dinner". You can talk about having dinner without mentioning the fact that you also ate it. Of course you ate it! The same is true with seeing a film. If you saw it, then of course you watched it.

Thank you so much for all your answers and explanations. 👍🙂