The condition is known as "strongly regular" (and I think that the modern usage is "regular semisimple", hence also "strongly regular semisimple", not just "regular", which is often understood to allow such things as regular unipotents).  By [Steinberg - Regular elements of semisimple algebraic groups](http://www.numdam.org/item?id=PMIHES_1965__25__49_0), say §§2.14, 2.15, the condition for a regular semisimple element $x$ to be strongly regular semisimple is that $x$ is regular semisimple, and the subgroup of the Weyl group $W(G, Z_G(x)^\circ)$ that fixes $x$ is trivial; and this extra condition is vacuous if $G$ is simply connected.

(Incidentally, in positive characteristic, a similar subtlety with possibly disconnected centralisers can arise even for regular semisimple elements of the Lie algebra.  For example, in characteristic $2$, the centraliser in $\operatorname{PGL}_2$ of the element $\operatorname{diag}(1, 0) \in \mathfrak{pgl}_2$ is the full normaliser of a torus.)