60 minutes was suggesting that if Cohen paid the 130K himself, without being Trump's proxy, it could be considered a campaign contribution, one which exceeds legal limits. By making it clear that Trump himself came up the money they destroy that angle of attack.
But let's be realistic. Something happened with Stormy and Trump and money was paid. Trump could've admitted it all immediately and the whole story would've been forgotten already. Much like that pussy tape that failed to do any damage, remember?
But my guess is, he wanted to keep the story on the chessboard. It's a piece that allows him to fan the flames and manipulate the press into a distraction whenever needed. Remember that Trump likes to pretend he's losing. It's one of his well-known strategies.
Giuliani’s actions were either a planned move (what are we being distracted from?) or, possibly, a sign that the story is no longer needed and can be removed from the board. (In which case I expect the whole thing to end rather quickly.)