Let $\cdot:G\times X \rightarrow X$ be a left group action. We know that:
$(i)$ The action is faithful, if $g\cdot x=x, \forall x\in X$ implies $g=e$;
$(ii)$ The action is free, if $g\cdot x=x$ for some $x\in X$ implies $g=e$;
$(iii)$ Every free action is faithful.
Thinking about the converse of $(iii)$, I realized that a faithful action, in order to be free, needs to have this property $(P)$: all stabilizer subgroups are equal (to the kernel of the action, of course). Thus, we have the equivalence: $\text{free} \Leftrightarrow \text{faithful}+(P)$.
Since I haven't found in literature any info about such actions with property $(P)$, I decided to call them quasi-free.
I am willing to find out any other properties of quasi-free actions. I have posted two answers, sharing my thoughts on this topic.