I am reading Hatcher's algebraic topology, where he defines a fibration $F\to E\to B$ to be principal if up to choices of homotopy equivalences it can be written as $\Omega B'\to F'\to E'\to B'$ (with B' being an extension of the fibration, $E'$ homotopy equivalent to $B$, $F'$ to $E$ and $\Omega B'$ to $F$) and such that all the maps and homotopy equivalences fit nicely into a commutative diagram.
However a principal bundle, according to wikipedia and my memory from an older course, is something different. A principal bundle being necessairly a principal $G$-bundle for some $G$. Which is defined as a fiber bundle such that each fiber has the structure of a $G$-torsor with some compatibility conditions.
My question is if we consider a principal fibration with Hatcher's definition, which happens to be a fiber bundle, is it then a principal bundle? Making Hatcher's use of the word principal a generelisation of a principal bundle. Or is the double use of the word principal a coincidence.
Thank you