Why has Saturn stopped contracting gravitationally?
The rate at which a planet contracts is determined by how much heat it can radiate from its surface and how close it is to the minimum radius allowed by electron degeneracy pressure (which is independent of temperature).
The contraction process is that heat radiates from the surface and is supplied by gravitational energy released by contraction. The process stops when the central pressure becomes determined by electron degeneracy at high densities. At this point the interior pressure decouples from the temperature (unlike a perfect gas) and the planet can cool at almost constant radius. The same process occurs in white dwarfs and brown dwarfs.
To first order, the radii of Jupiter and Saturn are similar and they have similar atmospheric compositions, so can radiate heat at a similar rate. But Saturn has far less gravitational potential energy to radiate and its centre reaches degenerate conditions earlier. In your equations you can see that $dR/dt \propto M^{-2}$ so Saturn contracts faster.
The equations you write down are essentially correct, but your statement that the terms are all non-zero is not. Effectively, Saturn is not luminous because of contraction, so $L \sim 0$ in these equations, and so is $dR/dt$ . The measured luminosity of Saturn instead comes from the residual heat in its interior and the chemical differentiation of helium and not through gravitational contraction.