The settlement agreement requires the payment to be made by either a cashier's check or money order. I made the payment with a personal check. The other party cashed the check and several days later called to thank me for the "Gift" and asked for his payment. What can I do about that?
You'd have to consult a lawyer in the state that the transaction took place to get a definitive answer. And also provide the details of the contract or settlement agreement.
That said, if you clearly presented the check as payment (verbally or otherwise) and they accepted and cashed the check, and it cleared, you should have good legal standing to force them to finalize the payment.
While they had every right to refuse the payment, and also every right to place a hold on the credit until the transaction cleared their bank, they don't have the right to simply claim the payment as a gift just because it came in a different form than they specified in the contract.
Obviously this is a lesson learned on reading the fine print though. And, to be frank, it sounds like someone wants to make life difficult for you for whatever reason. And if that is the case I would refer back to my initial comment about contacting a lawyer in that state.