Not sure I understand the first point, but if Trump doesn’t say it, it would not satisfy prosecutors.
As for the second, I’m not sure how this would help Cohen. He can always tell the truth himself that Trump paid him to make the thing go away. Most likely Rudy just assumed that is what Cohen is going to say and he’s just getting in front of it.
But again, I’m surprised it’s news at all. Everyone already knew these things.