In the Longman Dictionary, "Cup" is defined as "a sports competition in which a cup is given as a prize". So I think in the second sentence, "cup" means the competition itself.'Cup football' is the football that's played in competitions, like the FA Cup, or the World Cup'. The 'cup' is the prize that the winning team wins.
I can't imagine a context in which we would say 'football cup'.
Crossed with Rover. I agree, 'football cup' is the trophy.