Can you collect me from a private terminal near Key West?

Yes, and it is a different arrival from a commercial one. There is no baggage hall and no arrivals door, so the car meets the aircraft side of the operation rather than a kerb, and the timing follows the tail number rather than a published schedule.

Give the operator and the expected arrival when you book. Private arrivals move more than scheduled ones, sometimes by hours and sometimes by a day, and it is easier to hold a car against a moving time than to find one at short notice.

Onward from there is the ordinary service: one chauffeur, an agreed price, and a car that waits if the aircraft is early rather than being somewhere else. Around Key West that usually means Old Town, Duval Street, the waterfront or straight on to the next city.