Let us first assume that $G$ is a simple adjoint group and fix a maximal torus $T \leqslant G$. Let $W = N_G(T)/T$ be the corresponding Weyl group. What you're asking is, for a semisimple element $s \in T$, when does $W(s) = W$? Here $W(s) = \{w \in W \mid s^w = s\}$ is the centraliser of $s$ in $W$. As $G$ is adjoint this group is typically larger than the Weyl group of the reductive group $C_G(s)^{\circ}$.
Such a semisimple element is known as quasi-isolated, which means the group $C_G(s)$ is not contained in any proper Levi subgroup of $G$. This is equivalent to saying that $W(s)$ is not contained in any proper parabolic subgroup of $W$. Up to conjugacy such semisimple elements have been classified by Bonnafé in his truly beautiful article "Quasi-Isolated Elements in Reductive Groups", Communications in Algebra, 2005.
Outside of the identity element I think the ones you have given provide an exhaustive list.