I'm having a hard time understanding principal bundles. It seems there are a lot of definitions around which is making it even more confusing for me. Is this a valid definition of a principal bundle:
Fiber Bundle
A fiber bundle is a tuple $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: E \to B$ is a continuous surjective function called the projection. For each $b\in B$ there is a set $U \ni b$ such that $\pi^{-1}(U)$ is homeomorphic to $U\times F$ via homeomorphism $\varphi$ such that $\text{proj}_1 \circ \varphi = \pi$.
Principal Bundle
A principal bundle is a fiber bundle with fiber $F=G$ where $G$ is a topological group. The total space is denoted by $E = P$. Additionally we require a continuous right action of $G$ on $P$ denoted by $\lhd: P\times G \to P$. $\lhd$ preserves fibers so that if $p\in \pi^{-1}(b)$ with $b\in B$ then $p \lhd g \in \pi^{-1}(b)$ for $g\in G$. When we restrict $\lhd$ to fibers $\pi^{-1}(b)$ for $b\in B$ then $\lhd$ acts freely and transitively on $\pi^{-1}(b)$.
Are the above definitions correct? If not how can they be corrected? Are there other "better" (e.g. more useful or more intuitive) definitions that either are or aren't equivalent to these given?