Can you collect me from a private terminal near Miami Beach?

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 Miami Beach that usually means South Beach, Mid-Beach, North Beach or straight on to the next city.