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*](https://neil-strickland.staff.shef.ac.uk/research/jordan.pdf) (see also [unofficial errata and details filled in](http://www.cip.ifi.lmu.de/~grinberg/algebra/jordan-errata.pdf)), 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 $V$).