In the next sentence:
"We use the Internet to watch videos, play games, or watch movies"
Why we say watch movies and not see movies?

See a translation

You could say "see", but "watch" is more common in this context. As far as I know there is no logic to it, it's just something you have to memorize.

I saw 2 movies yesterday = I watched two movies yesterday. Both sound fine. In my opinion it doesn't matter whether it's at home or at a theatre.

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In many cases the difference is that 'watch' is about the intention and 'see' is about the result. This does not apply to your example though.

I saw an accident; I 'watched' an accident sounds a little odd because usually you 'see' an accident without meaning to.

I like to watch my kids play soccer; I 'see' sounds a little odd because usually watching your kids is something you want to do.