The same load in the Z direction is applied to each point - but that’s not a height - it is the direction and strength the point is being pulled with.
For a simpler example think of a single chain, fixed at either end, hanging under gravity to form a catenary curve. The mass of the chain (and therefore the load) is evenly distributed along its length, but of course the middle part ends up lower.
The strength of the load here just affects how much the springs will stretch.
If you want to model something like a Gaudi style hanging chain model, where the stretch is very small, make the springs very stiff (but longer than their flat length, otherwise it won’t be able to move).
As you tend towards inextensible cables, the actual weight used becomes less and less important - just as a steel cable or a nylon cable of the same length will both form approximately the same catenary shape when hanging under their own weight (assuming they are long enough that bending resistance doesn’t play a large part, and the strength to weight ratio of both materials is high).