Going back towards the original point, there seems to be a degree of participation or involvement when we ‘see’ but watching is more voyeuristic and dispassionate.

“Did you see that?” implies that the spectacle should have evoked a response in you.

“I watched it happen.” Suggests a deliberate standing back whereas “I saw it happen” does not have that connotation.

This may explain why we go to see a film but watch the TV. There is more deliberate action and effort when we go to the cinema or the theatre, and the whole experience is more involving than when we just sit at home and watch TV.