In North America, the majority of people only use the word theater to encompass both the movie and live stage presentations. The word "cinema" is actually French but adopted into English because of marketing moguls. Cinema means movie or кіно, but due to evolution of language is the common understanding of "salle de cinema" . Having said that, it is very common for people to say "I am at (or going to) the movies" to really be saying "I am at (or going to) the movie theater" There are many variants on English in conveying a particular message. I think this is true of українсьсу as well.
You are bringing up many irrelevant things.
(1) The relevant thing: do people use the word "theater" (without the word "movie" or "movies") to mean "movie theater"? If somebody says "I'm going to the theater", can it also mean that they are actually going to the movie theater, not a normal one?
Also, even if it is used like that, I would not accept it here. Accepting "theater" as a translation means I won't be able to differentiate between two cases: 1) the learner actually knows what "кінотеатр" means and they just used "theater" because it can also mean "movie theater"; 2) the learner thinks "кінотеатр" means "theater", as in, simply a theater.
(2) Irrelevant: cinema can also mean "movie theater". Both are accepted here.
(3) Irrelevant: "I am at the movies" = "I am at the movie theather", "I am going to the movies = I am going to the movie theater". Yes. But it has nothing to do with the sentence "This is a very big cinema/movie theater". How is it relevant? You can't say "This is a very big movies". The phrase "at the movies" that you brought up is irrelevant. But yes, it's a thing, and it's accepted in other relevant exercises.
I appreciate the difficulties of making sure that people understand the specific meaning of the words they translate (MOVIE theater), but Americans usually say "theater" for the meaning 'movie theater'. In fact, when a new film is being advertised, the voice-over or the label will always say "in theaters now" rather than "in movie theaters now." Its not a big deal, though.