Definition of polycyclic groups
Confusion arises because the same term normal series is used differently in the literature. To also quote Wikipedia
If in addition each $A_i$ is normal in $G$, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series.
So there are at least two schools:
- Call the weaker case subnormal and the stronger case normal series
- Call the weaker case normal and the stronger case invariant series.
As a side-effect of the multi-authorship, it is perhaps not completely enforceable that Wikipedia agree upon one of the two schemes consistently (perhaps it would be best to use only subnormal and invariant and get rid of the ambiguous normal)
The link you provided is for supersolvable groups. If you look over at solvable groups, they handle the issue of equivalence of some definitions. Here's a link:"Solvable group - Wikipedia" https://en.m.wikipedia.org/wiki/Solvable_group
If this is not the issue your having, I think the answer lies in the fact that they explicitly say that polycyclic groups are solvable. So I don't really see what the problem is. Afterwards they tack on some additional things, but unless you can show there's a contradiction, it's fine.