You are and were absolutely correct, OZzTYn.
Yes, it is most frustrating.
You must be really frustrated about that.
Let's go to the movies....................you may not have a specific movie in mind.
Shall we go to a movie..........you may have a specific movie in mind,
So, in a general sense, they have the same meaning.
In British English we say, "Let's go to the cinema", and that takes care of both 'go to a movie' and 'go to the movies'