A river system just needs the illusion of moving water is absolutly enough and this is how it works in most of the games. Just one interaction is important: If you are in a river, you move with the river and the moving is indicated with some visible effects at the river rim.
Varying water is really important for the game:
The river starts at a spring on a mountain, which could comes from a cave or from a small gap in some rocks and starts as small creek and gets bigger and bigger. Water from the river you can drink. The river bed is made of round stones(gravel)
There could be also greenish ponds in the moderate bioms with reed, water lilies, dragonflies and frogs. You cannot drink from ponds without boiling the water at a fire.
In the savana are brown water holes. Also boil it before drinking it.
In the snowy areas are frozen lakes and there could be water below the ice which you can also drink directly.
ozean water is not drinkable and increases thirst.
How is the river changing if you terraform? In the same way as water ponds: There is just a particel effect of spray&foam if the movement goes to a death end.
The steepness of the riverbed changes the speed of the river and the animation of the river movement from slow moving water to a waterfall.