> I am baffled by the definition of $S_\pi$ since to me it seems as if there is always only one element contained in $S_\pi$, namely the subspace spanned by the eigenvectors corresponding to the eigenvalues $\lambda_{\ell_j}$, $j = 1,\ldots,r$, of $A$.

This logic is wrong. Try the simplest nontrivial case -- the case when $n=2$ and $r=1$ and $\pi=\left(0,1\right)$. Then, $S_\pi$ contains all $1$-dimensional subspaces $V$ of the $2$-dimensional vector space $\mathbb{C}^n$ satisfying $V \cap A_1 = 0$ (and $\dim\left(V \cap A_2\right) = 1$, but this is automatically satisfied because $A_2 = \mathbb{C}^n$ and $V$ is $1$-dimensional). These are all $1$-dimensional subspaces of $\mathbb{C}^n$ except for $A_1$ itself (which lies in $S_{\left(1,0\right)}$ instead).