Though a lot of work on this topic still should be done, I hoped that the basic question has already been solved. To my surprize, I see that this is not true. Let me very briefly touch these questions again - I do not claim to give complete answer. However, the basic distinction seems to me clear, and the work to be done should especially clarify the details. Let me stress that in general, X is not a set.
Fuzzy set is then characterization of X which uses some scale.
In vague characterization of grouping, time plays no role. Yes, we need probability to describe them but fuzziness as well. Their probabilistic interpretation is forced. This is nonsense. Let all of us better study both these phenomena and try to find their interconnection which surely is very deeply hidden somewhere.