Usage of radiator as a device for engine cooling

The word radiator is

a hollow metal device for heating rooms. Radiators are usually connected by pipes through which hot water is sent

But it also has the meaning of

a device for cooling the engine of a vehicle or an aircraft

which is the exact opposite.

Etymology says

1836, "any thing that radiates," agent noun in Latin form from radiate. Meaning "heater" is from 1851; sense of "cooling device in internal combustion engine" is 1900.

But I can't see why radiator is used for a cooling device, too. What is the reason?

Interestingly, since 1900 the use of radiator is massively increased:

Solution 1:

They work in exactly the same way - take heat from the circulating water, and radiate it.

The only difference is that in one case you want the heat in the room, and in the other you want to dissipate it to the surrounding air.