I have two definition of $G$-principal fiber bundle when $G$ is a linear algebraic group complex. Let be $X$ a complex variety.
A principal fiber bundle on $X$ is a couple ($\xi,q$) where $\xi$ is a complex variety and $q: \xi\rightarrow X$ a morphism such that:
i) there is an action of G on $\xi$ such that the fiber $q^{-1}(X)$ are fixed
ii) is locally trivial: for each $x \in X$ exists $x \in U \subset X$ neighborhood of $x$ in $X$ and am isomorphism $\psi_{U}:q^{-1}(U) \rightarrow G \times U$ such that $\psi_{U}$ is $G$-equviariant
the $G$ action is free and transitive
The second definition is as the first but without the third axiom, so I would like to know if it's a real axiom and if there are some examples of principal fiber bundle which are graphically significative