Again, perhaps it's a difference between BE and AE. I'd take the museum sentence to refer to the coming Saturday/Sunday. If I were talking about an event far into the future (or the past) I'd probably say "the Sunday". "We're going to my sister's for Christmas and then we're going to my brother's on the Sunday", i.e. the Sunday after Christmas.
I agree that "a Sunday" is non-specific; we don't know which Sunday. "We'll need to visit him on a Sunday" - we're only visiting him once. However, I suppose you could say "Jews go to the synagogue on a Saturday" - on Saturdays in general.