Translation:I went to a movie theater with a neighbor.
Really?.. I'm not a native speaker but just logically "I'm going to the theater" does sound like going to the actual theater... Or is it like young people don't go to theaters anymore so it's obvious that it's a movie theater and not a theater theater :D
It depends a little on the context, but I've heard many people say that. It's a bit colloquial, though.
I don't think I ever heard people refer to movie theater simply as "theater", at least where I lived in the the coastal states... Might be some rare regional thing
Well, different regions, different habits. Say, in Ukrainian we always drop the "theatre" part actually, we say just "Я ходила в кіно", I can't remember last time I actually said the full word "кінотеатр" :)
According to the COCA, "theater" meaning "cinema" accounts to ~80% of the use in the USA, so... not so rare!
I have been living in the USA only for a handful of months now, and I'm not a native speaker of English, but I have been aware of this use for many years.
Well, it comes from movie theater of course, and I can imagine it's more common for some older generation, but it was just "to the theater" - I can't imagine Americans saying "movie theater" all the time, when they have the tendency to shorten all the words as much as possible!
Lately I only read "to the movies", as you wrote :)