//Does it make a difference that the obstruction is below the surface level of the river, creating the continuous wave above? //
Not my field of expertise but I would venture to say 'no difference'.
If we picture a block of stone, tall enough to stand above water level, the water would still heap up against it, forming the standing wave. Assuming the river flow is not strong enough to defy gravity and flow up and over the top of the object, then the water has one less degree of freedom of movement and must flow around the sides, which would mean more foaming but otherwise not prevent the formation of the standing wave.
//and what would you describe then as a standing wave.//
Again, not my field of expertise but, in common with those lenticular clouds above mountaintops the features are fluid flow combined with vertical displacement of the flow by an obstruction.