"Directory" for the main board in an airport, etc., informing people on arrivals/departures, floors/levels to certain stores, etc

Solution 1:

You've actually asked about several different types of signage in your question.

If you are asking about a building with a sign directing you to various floors or offices, then this is called a building directory. Oddly, I cannot find a dictionary definition for this use either. But, here is a company that makes electronic versions of them.

If you are speaking of a sign that tells airport departure or arrival times, gates, flight info, this is called a board. (e.g. Departure Board) You could make a case to refer to this as a directory, but most airports call this the board or information board.

If you are speaking of a board that tells you theatre times, then this is called a theatre schedule board. I don't know that this is a formal name for it, but that's what I've always called it.

And, for anything that hangs on a wall or for a ceiling that tells you where you are going, you can always use the catch-all sign to describe this.