Salmonee,
The radiator confusion may have come from an analogy I made comparing a fresh water cooled engine's "heat exchanger" with a radiator. The heat exchanger is where the "closed" or "fresh water" part of the cooling system (surrounding the engine) is cooled by the lake or sea water (often called raw water).
With a typical radiator the engine coolant is cooled by ambient air which is never seen on a small boat. The air is drawn across the radiator either by a fan or vehicle motion or both. In the case of a marine engine with fresh water cooling, the closed system's heat is cooled by ambient water that is drawn up from the lake or ocean by the "raw water pump" and pumped into a fairly large cylinder that is about 4 - 6 inches in diameter and 12 -18 inches wide and it's called a "heat exchanger", from there the raw water usually cools the manifolds and exhaust and then exits back to the lake. Inside the heat exchanger there is a series of tubes that contains the engine coolant which is pumped by the engine's traditional water pump. This pump is usually called the circulating pump as it simply circulates coolant throughout the engine and through some sort of heat exchanger. On your car that heat exchanger is called a radiator (see above) and on a boat it is called a heat exchanger oddly enough . . .Both could technically be referred to as radiators as they both radiate heat, or heat exchangers as they both exchange heat. One from engine to air, and the other from engine to "raw water". Hence the confusion.
The raw water pump is the one with the rubber impeller. It can be located in the drive in the case of a Mercruiser Alpha and most OMC drives, or the raw water pump can be mounted on the front of the engine in the case of a Volvo or Mercruiser Bravo, or a traditional inboard. In all of these cases, the rubber impeller would be damaged very quickly by running it out of the water without some sort of water supply directly to the raw water pump such as "muffs". Muffs are a set of rubber thingees that fit over an outdrive's water intake grates and allow a garden hose to supply water similar to what would happen if those grates were under water . . .