Incredible River Under The Sea!

The article in the OP makes some unusual claims, and surprise surprise, it's not quite accurate.
The article in the OP conflates two remarkably different things.

1) A recent discovery of an undersea river.
2) A sinkhole in Mexico with a superficial resemblance to a river.

The images and videos of scuba divers in an apparent underwater river are from Cenote Angelita. A cenote is a natural pit - a cave or sinkhole. If you watch the videos of this same area, rather than subscribe to the bizarre notions of the website in the OP, you can see that this is - essentially - in a sinkhole. It might appear to be a river, but it's not moving. There is a layer of slightly acidic (and highly toxic if you were to breathe it) hydrogen sulfate separating two layers of water: freshwater above, and saltwater below. Water with more salt content is denser, and in relatively undisturbed areas (like this sinkhole) freshwater will not mix much with saltwater and will instead remain on top of the saltwater.

The Black Sea undersea river is an entirely different thing. For one, it's actually moving. I don't know that there are any images of it aside from computer models. Part of the reason for this is because much of the upper depths of the Black Sea is brackish water - a classification of water which has a salinity between freshwater and saltwater. It works this way because it is a density current - a river of even denser seawater flowing into the Black Sea from the Bosphorus Strait. It carries sediment and nutrients along its length, as well as having a much higher salinity. Recall that higher salinity will be below lower salinity from earlier - this is the case here as well. Here, there is probably mixing throughout the length of the river, but the speed and volume at which this river moves and the fact that the Black Sea also empties its upper (less dense) waters back through the Bosphorus Strait separates it somewhat. The colder, less-saline (brackish) upper waters of the Black Sea move towards the Mediterranean through this straight, while the denser, warmer waters of the deep Mediterranean move towards the Black Sea as a two-way exchange. This also leads to very low oxygen levels in the deeper parts of the Black Sea.