Bingo.
It’s a binding contract, exactly as you said. It does not have to be written, and if written, it does not have to be signed by either party.
In fact, Cohen, the attorney, ay not have even revealed the arrangement to Trump. The contract may be between Cohen and Daniels, just as binding.
When one party accepts the money and makes consideration, that is performance and proof of a contract.
Now, here’s a question, did she declare and pay taxes on the money?
I don't care if Trump had sex with a slut ten or twenty years ago... or if he had sex with a hundred sluts...