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 $V$).