The idea that vectors representing angular momenta are precessing is "semi-classical", i.e., supposed to be an aid in understanding what's going on. In reality such pictures of the "vector model" sometimes serve only to add confusion.
As we know, the rules for angular momenta arise from corresponding operators and their group theory. The truth about $L-S$ coupling or $jj$ coupling is that in lighter atoms the Hamiltonian ($H$) is such that the individual electrons' total angular momenta $j_i$ are not good* quantum numbers, and it is better to use instead the total orbital angular momentum ($L^2$) and the total spin ($S^2$) to describe the atom. In the semi-classical model we would say that individual $\vec l_i$ "precess" around $\vec L$ (and similarly for $\vec s_i$ and $\vec S$). [Note that $S$ enters the picture indirectly due to anti-symmetry of the overall atomic state under electron exchange: the total spin $S$-state must be combined with the (conserved) total $L$-state to be antisymmetric under electron exchange.]
For heavy atoms the spin-orbit or $jj$ coupling dominates, i.e., the total angular momenta $j_i$ of individual electrons are good quantum numbers due to the $\vec l_i\cdot\vec s_i$ terms. Thus the analogy here is that the individual $\vec j_i$ "precess" around $\vec J$.
*Good quantum numbers are eigenvalues of operators that commute with the Hamiltonian so that the corresponding physical quantities are constant in time and thus these quantum numbers can be used to label states.
P.S. It is easy to mistake the term $L-S$ coupling for spin-orbit coupling, but it is really $jj$ coupling that is spin-orbit coupling.