Unfortunately, the real world doesn't fit neatly into categories, and sometimes neither do our imitations of it. Currently, the terms terrarium, paludarium, vivarium, and occasionally riparium are used more or less interchangeably without any kind of definition or standard:
"terrarium" even if there's a very large amount of water or no living plants.
"paludarium" usually for habitat which isn't the least bit swampy.
"vivarium" for any cage with animals and not necessarily any live plants.
"riparium" usually for an aquarium with 'extra' circulation or filtration.

The last one at least comes close, although it doesn't otherwise reflect an identifiable river or stream habitat.