Yes, this definition is quite clearly wrong. The sum of the $\mathrm{dim}(V \cap A_i)/(V \cap A_{i-1})$ over all $i = 1, 2, \ldots, n$ is $\dim V - \dim 0 = n$, whereas the sum of the $\pi\left(i\right)$ is $1+2+\cdots+n$ which is usually larger.

I suspect that what the author wanted to say is somewhere in §4 of Neil Strickland, The Steinberg module and the Hecke algebra (see also unofficial errata and details filled in), except that Strickland uses the standard basis for what the authors of your papers use the eigenbasis of $A$ (but this matters little, since you can turn every basis of $V$ into the standard basis by an automorphism of $V$).

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).