Ordinary Weyl orbits do not classify central characters

The dot action gives $s\cdot 0=s(\rho )-\rho =-2\omega =-\alpha$. Since $⟨\rho ,{\alpha }^{\vee }⟩=1$, the singular-vector lemma The simple-root singular vector in a Verma module gives a nonzero highest vector $f{v}_0$ of weight $-2\omega$ in $M(0)$. By The universal property of Verma modules it induces a nonzero map $M(-2\omega )\to M(0)$. Every central element acts by a scalar on either Verma module by The Harish-Chandra projection computes the highest-weight scalar; the nonzero map intertwines these actions, so their central characters are equal.

Under the ordinary Weyl action, $s(0)=0$, so the ordinary orbit of $0$ is just $\{0\}$, while the orbit of $-2\omega$ is $\{\pm 2\omega \}$. Thus the two weights are not ordinarily conjugate.

Therefore ordinary Weyl orbits are too fine here: they separate weights that have the same central character. The $\rho$-shifted dot action is essential.