When you click on a curve and move it: it's a point of this curve, and when you move it you have some labels on snaped surface for help you!

But post a little file 3dm of your problem!



ps I don't know what do you want make with your curves on the surface but maybe this can help you!
It's not points of curve on the surface but adding control points of the surface!
You can of course call the Add Points function several times!

Surface must be selected before calling the Function!



Now you can move any control points and the surface will follow them! ;)

A cool sort of organic modeling with nurbs! :)