The basic principle is simple: You have TWO terrains, one is a duplicate of the original and cut away to leave the river behind, thus, it follows the exact same height and slope as the original terrain, which real rivers do!

1. Edit the main terrain for the landscape
2. Dig out where the river chanel will be with the "Dig" tool, but, you don't want it very deep at all.
3. Duplicate this terrain and edit it
4. In the terrain editor, cut away the land which will not be the river
5. Move the river vertical down very slightly as it would be in real life.
6. Smooth the river terrain if necessary to remove excess roughness.
7. Apply water material.

1. In the terrain editor, save the terrain greyscale out as a 16 bit tif image. You need 16 bit images to avoid the "stair stepping" caused by the fact that 8 bit images only allow for 256 shades of grey, so that's 256 height differences which is bad for precision...but you can live with it if you have no choice, using some "erosion" to hide it.
2. Open in Photoshop or similar app that can open 16 bit tif's (not all can, if not as said, no big deal, you can use some gentle erosion later to hide it)
3. make a new layer for the river shape
4. paint in black, the shape of the river you want.
5. Make this layer "Soft Light", it should now darken slightly the main terrain layer below it. Remember, dark = lower, in greyscales for terrains! What we're doing is using a line or pianted image, to smoothly lower underlying terrain as a gradient, rather than just dig it down to all the same flat height.
6. Save this image out as 16 bit tif
7. Make a selection based on the river layout itself and make a new layer from this selection, so you have both the river and original terrain layer underneath combined. This layer will just have the heightfield of the river itself.
8. Make a new layer and fill it with black. Move this layer underneath the river heightfield layer.
9. Save the heightfield and black layer out as a new image so this will give you the river terrain heightfield.
   Apply this image to the river terrain

HERE, THE RIVER LINE AND TERRAIN UNDEREATH IS SELECTED, AND COPIED TO A NEW LAYER,
AND A LAYER WITH BLACK PLACED UNDERNEATH. THIS WILL SHAPE THE RIVER TERRAIN.

TERRAIN WITH THE NEW GREYSCALE APPLIED
(plus an earlier bit to the right corner I dug out for a different attempt at hand sculpting this, lol)

THIS IS THE RIVER TERRAIN

NOW, BOTH ARE SHOWING
YOU CAN SEE THE RIVER FOLLOWS THE SAME HEIGHT AS THE TERRAIN!