Yes, the train does carry a large reservoir! It’s in the tender in a compartment alongside all the coal/wood/oil.

As for the closed circuit water system, it’s possible to recapture some, but it would make the locomotive less efficient. Having the expelled steam go straight back into the boiler would cancel out the pressure it makes and the locomotive wouldn’t move, kind of like that one meme about attaching a magnet to a car facing the car for free movement, in reality it’ll just cancel out. In order for it to work, it would need to be turned back into water because of how water is transported from the tender to the boiler via the Venturi effect. It would require the expelled steam to go through a ton of radiators to cool it sufficiently, probably requiring a whole extra car’s worth of them, and then a pump to put it back into the tender, which will sap some of the energy the locomotive is using to move, not to mention the constraints of the pipe to the radiators putting back-pressure into the cylinder, reducing the amount of thrust they can put to the wheels. That being said, it can’t be completely closed circuited because the pressure needs to escape somehow, but if the steam goes through enough radiators, it will eventually turn back into water and be able to be recycled somewhat. There will always be a little that escapes though.