As Nanren said, "the movie" doesn't really mean the physical place the movie is shown. In this kind of situation, "the movie" can refer to the physical picture in front of you, as well as the event. So if you are going to the shopping mall, you would never say "there's a movie at the mall", meaning a theater that shows movies. But you do use "the movie" to refer to the event of seeing the film as well as the actual picture itself. A lot of words can work this way. "Did you make it to the football game yesterday?": Here, "the football game" means the event, which takes place in a stadium. "Let's go to dinner!": "Dinner" means the event of eating dinner, not the physical restaurant you're going to. "Where's the party?" & "The party is fun.": Party is used the same way both times, in both cases you mean the event.

A 'movie theater'/ 'cinema' is the building where people go to watch a movie/ a film. In your example, the use of 'movie' in both cases refers to the film that you are watching. If instead, the example was: "Did you make it to the movie theater on time?" "Yeah, just as I sat down, the movie started." The first use of 'movie' is attached to 'movie theater' so it is referring to the building you would go to, to watch the film. In the second use, 'movie' is referring to the 'film.' I hope this helps!

A movie theatre is a theatre where movies are shown. "movie" is an adjective, it describes the theatre. There are also 'concert theatres', 'music theatres', opera theatre, etc.

Thank you Anna

Matt's answer is exactly right. One usage note I would add is that where I live we do say, "Let's go to a movie" or "We went to the movies." Here it doesn't refer to the building, but like James said, the event or experience.