How does ventilation work in a steam room?

My local leisure centre has a steam room just off of the swimming pool. This has a fan in the ceiling that blows cooler air from outside into the room that then, via the steam generator input, turn into lovely blankets of steam, that then exit under the door into the pool area.

I’ve been told that the usual way the ventilation works in these things is the other way round, i.e. with the fan extracting and cool air entering under the door. But, because the pool area is at a high-ish temperature this wouldn’t create enough steam, so the fan blows.

My question is, is this correct?

I tried googling. I can’t really find much, but there’s one site that says “The air outlet must always lead into the same room from which the inlet air is drawn.” Why would this be the case?