In my opinion there is no vagueness to the sense of the word. The meaning of the word "choice" is identical both in "You have a choice" and "You have two choices".
This is because the choice in "You have a choice" is the one that you choose. Sure, there may be a variety of options available, but you only ever have the one that you end up choosing.
This, however, does not break contract of meaning with the sentence "You have two choices". Sure, there is a supplement of information concerning the perceived number of options available, which analogously insinuates the reality of multiple simultaneously choosable options, but they will only ever be options to choose from singularly.