Can you collect me from a private terminal?

Yes, and it is a different arrival from a commercial one. There is no baggage hall and no arrivals door, so the car meets you where the FBO allows, usually at its front door, or planeside with the operator's approval, and the timing follows the operator's arrival time 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, anywhere in Miami, Miami Beach, Fort Lauderdale, Palm Beach, Boca Raton, Wellington, Naples, Key West, Orlando and Tampa or straight on to the next city.