Okay, so I thought back to my physics days some more, and the real understanding of the problem lies in understanding the definition of tension. The scale is ac

Okay, so I thought back to my physics days some more, and the real understanding of the problem lies in understanding the definition of tension. The scale is actually measuring tension in the rope. An informal definition of tension: cut the rope at any point, and determine what force needs to be applied to the one of the cut ends to keep that part of the rope in equilibrium. That force is the tension in the rope. So in the second example, just measure the tension in the rope.