Extrude curve into open Mesh

I have large open meshes representing mountains/terrain. Like what’s available from Terrain2STL.

I carve those into wood with a CNC machine, but I can’t integrate the geometry of the bodies of water into the Mesh geometry, so I have to machine the bodies of water in a lengthy second process. I’m seeking advice on how to go about stamping the bodies of water into the mesh directly to eliminate the second process.

I have Closed curves representing the bodies of water that I would like to project onto the mesh, then extrude the shape of the curve into the mesh, ideally resulting in a flat surface stamped “some depth” into the mesh. This will leave a nice recessed outline of the bodies of water for painting.

I am a novice user of Rhino, but I have worked in CAD software for a long time. Anything you can offer would be greatly appreciated. Thanks!

Hello - please post a simple example file.