Let $Z(\mathfrak{g})$ be the centre of $U(\mathfrak{g})$ and let $\chi_\lambda$ be an algebra homomorphism $Z(\mathfrak{g}) \to \mathbb{C}$ such that $ z \cdot v = \chi_\lambda(z)v $ for all $z \in Z(\mathfrak{g})$, $v \in M(\lambda)$, then $M_I(\lambda)$ and its subquotients including $L(\lambda)$ have the same infinitesimal character $\chi_\lambda$.

Why we call the infinitesimal character $\chi_\lambda$ of $L(\lambda)$ regular if $\langle \lambda + \rho,\alpha^\lor\rangle \neq 0$ for all $\alpha\in\Phi$ instead of $\langle \lambda,\alpha^\lor\rangle \neq 0$ for all $\alpha\in\Phi$?

I know that $\chi_\lambda=\chi_\mu$ iff $\lambda=w\cdot\mu$ for some $w\in W$. It makes more sense to define in the former way. But is there any other/fundamental reason for the definition?