isn't "Vice city" just a nickname for Miami? I mean The City of "Vice" is in Florida (real state, obviously) but on the logo it is not Vice City but Vice city - so maybe it was just a nickname for Miami (but they changed the name of the city later) and the logo remained unchanged? Look:
It is not Vice City but VIce city, and vice = bad habit. So maybe it was GTA: City of bad habits (Miami).