I've finally gotten around to learning about principal $G$-bundles.
In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether these definitions are equivalent or not. I would appreciate any clarification.
All maps and group actions are assumed continuous.
Definition 1: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:
(1) $G$ acts freely and transitively on fibers.
(2A) $G$ preserves fibers.
Definition 2: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a left action of $G$ on $F$ (note $F$ here) such that:
(1) $G$ acts freely and transitively on $F$.
(2B) There exists a trivializing cover with $G$-valued transition maps.
Definition 3: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:
(1') $G$ acts freely on $P$ and $X = P/G$ and $\pi\colon P \to X$ is $p \mapsto [p]$.
(2C) There exists a trivializing cover that is $G$-equivariant.
Definition 4: A principal $G$-bundle is a fiber bundle $F \to P \xrightarrow{\pi} X$ together with a right action of $G$ on $P$ such that:
(2A) $G$ preserves fibers.
(2C) There exists a trivializing cover that is $G$-equivariant.
Thoughts: It seems to me that Definition 4 is not equivalent to the other three. More than anything else, I am unclear as to why the existence of a trivializing cover that is $G$-equivariant is equivalent (is it?) to the existence of one that has $G$-valued transition functions.
I've also seen a fifth definition which assumes only condition (1).
Thanks in advance.