Just to be pedantic, nuclear plants generate electricity at about 33% efficiency. So to get 1000 MW of electricity, you start with about 3000 MW thermal.
Turns out 1% of three gigawatts is even more. :)
It's definitely a lot. But nothing naturally-circulating sodium metal through 4 redundant NaK-air dump heat exchangers can't handle.
While there are some pure air cooled engines, lawn mowers, motor cycles, etc. A typical car engine is also considered air cooled since it gives off its heat to atmosphere even with assistance of a coolant radiator system.
Most marine engines, some power plants and some industrial equipment is considered liquid cooled since the coolant passes through a fluid to fluid heat exchanger and then that cooling water gets put back into the ocean/river. An example would be some ABB drives come in a LC liquid cooled or AC air cooled package. The motor drive lineups are the same with primary coolant going through all the drive bays the difference is at the end if they put an liquid to air or liquid to liquid heat exchanger.
We just have different definitions then. In car circles your typical car is definitely water cooled. I had never seen your definition but it also makes sense. The typical one is describing what fluid contacts the hot parts and yours is describing what mass the heat is transferred to.
My late 90s era Ducati is considered my most people to be "air cooled", but it has oil galleries that circle around the cylinder walls and an air-oil radiator. (Pedants might call it "oil cooled", but those are mostly the same people who "well actually" you when you call it a V Twin. "Well, actually, it's an _L_ Twin..." - just because it's a 90 degree V Twin rolled forward so the front cylinder is roughly horizontal and the read is mostly vertical...)
(And to further complicate matters, the 900cc two valve "air cooled" Ducati motors have those cylinder cooling oil galleries, but the 750cc motors also have oil coolers, but do not circulate the oil in the cylinder walls. The 600cc motors use the same oil galleries as the 750s, but do not come with the oil cooler...)