Courier or carrier?

All of them carry stuff, and can therefore be called carriers, as is true of trucking companies, airlines, and so forth. When arranging for shipment of goods with someone, one may thus ask ‘Which carrier will you use for the shipment?’ Possible answers to that question include all of the companies mentioned by the OP.

While some definitions of the word courier make it appear interchangeable with carrier, its typical contemporary usage is limited to the companies that claim to provide faster, better service than the mail; it thus excludes the postal service. Asking someone ‘Which courier will you use for the shipment?’ will generally be understood to imply that one expects the answer to be something like UPS or FedEx, and not USPS. (Incidentally, UPS, FedEx, and suchlike would not normally be called postal services; that term is reserved for the USPS and its counterparts in other countries.)