  In these notes, we obtain new stability estimates for centered
self-decomposable probability measures on $\mathbb{R}^d$ with finite second
moment and for non-degenerate symmetric $\alpha$-stable probability measures on
$\mathbb{R}^d$ with $\alpha \in (1,2)$. These new results are refinements of
the corresponding ones available in the literature. The proofs are based on
Stein's method for self-decomposable laws, recently developed in a series of
papers, and on closed forms techniques together with a new ingredient: weighted
Poincar\'e-type inequalities. As applications, rates of convergence in
Wasserstein-type distances are computed for several instances of the
generalized central limit theorems (CLTs). In particular, a
$n^{1-2/\alpha}$-rate is obtained in $1$-Wasserstein distance when the target
law is a non-degenerate symmetric $\alpha$-stable one with $\alpha \in (1,2)$.
Finally, the non-degenerate symmetric Cauchy case is studied at length from a
spectral point of view. At last, in this Cauchy situation, a $n^{-1}$-rate of
convergence is obtained when the initial law is a certain instance of layered
stable distributions.
