I have 2 definition of fibre bundles, one from the book Fibre bundles from Husemoller and the other one from The topology of fibre bundles by Steenrod. And they differ.
The first definition says that if $(X,p,B)$ is a principal G-bundle (that is, "a bundle" of the form $(X,pr, X \mod G)$, where $G$ has free action on $X$ and $pr$ is canonical projection), F is a right G-space (that is, $F$ has free right action of G) then fibre bundle is triple $(X \times F \mod G, p_{F}, B)$, where $G$ has action on $X \times F$ of the form $(x,y)s=(xs,s^{-1}y)$, and $p_{F}$ is factorization of projection to the first coordinate composed with $p$.
The second definiton first introduces coordinate bundle that contains the general "bundle" $(X,p,B)$ with fibre $Y$ and with group $G$ having effective action on $Y$. The definition requires local triviality $\phi_{j}:V_{j}\times Y \to p^{-1}(V_{j})$, where $V_{j}$ is covering of $B$. And requires coordinate transformations between $\phi_{i},\phi_{j}$ on common space $V_{i}\cap V_{j}$ coincides with the operation of an element $g \in G$ (meaning I guess that it's a map $y \mapsto g \times y$ on $Y$ for some $g$) Fibre bundle can be then understood as being maximal with respect to the amount of coordinate transformation functions (roughly speaking...)
My question is, how much do these definitions differ? Is the first definition more strict when we consider locally trivial fibre bundles? Also why does one of them need free action and the other one effective action?