Medallion cabs run mostly in lower and midtown Manhattan, but can pickup anywhere in the city. Medallions also routinely cost over $500,000 to obtain. Medallion cabs are tightly controlled and regulated.
Boro cabs, from summer 2013, can pickup in upper Manhattan or any of the outer boroughs, but must stay out of the airports. Along with strictly car services, they are called "livery cabs" because of the way they are regulated, and before 2013, were limited to Call and Demand procuring of trips. A car service would have been appropriate for Mr. Simon.
The last category of cab in NYC is Uber/Lyft, whatever they are.