That isn't how I would understand it. I would say that normally they all meant the same thing.
'flip the card ' sounds slightly more informal than 'turn over the card' ('turn the card over' is another possibility).
I would have thought that 'turn the card face up' is perhaps more usual than 'turn the card up'
But, honestly, I can't see a great difference between any of your expressions and they alll sound fine to me.
Depending on the type of card game being played and what was going on at the moment a person said any of the three, they might all mean the same thing or they might mean different things, exactly in the way CarlosRapido explained. All the same, an English speaker presented with these phrases in the absence of context would probably suggest they all mean the same. That is, however, my guess at probabilities.