What is a G-covering/trivial covering?

I have a problem understanding G-coverings.

If $G$ is a group and $Y$ a topological space, then one can define an action $$A:G\times Y\rightarrow Y;\,\,\,(g,y)\mapsto g\cdot y$$ We can thefine an equivalence relation on Y $\sim_y$ as follows $$ y\sim_y y' \Leftrightarrow \exists g\in G: A(g,y)=y' \Leftrightarrow g\cdot y=y'$$ and so there is also the set $$Y/\sim_y:=Y/G=\{[y]|y\in Y\}$$ where $$[y]=\{g\cdot y|g\in G\}$$ Now we can look at the projection map $$p:Y\rightarrow Y/G; \,\,\,y\mapsto [y]$$ which is continuous since $Y/G$ is endowed the the quotient topology.

Is this correct so far?

We have also seen the notion of acting evenly on $Y$ and defined a $G$-covering map as the projection $p$ when $G$ acts evenly on $Y$, i.e. the action $A$ is even. But then we also saw the notion of the trivial $G$-covering on $Y/G$ but there I somehow don't see it.

The trivial $G$-covering of $Y/G$ is the product $$Y/G\times G\rightarrow Y/G$$ where G acts by (left) mulpiplication on the second factor.

That's our definition. But somehow I don't see the connection from this to the definition of a $G$ covering. Since there we don't have a product, we only consider the projection map. Could someone help me?

Thanks a lot.

A correct replacement for the highlighted sentence is to say for instance:

Take a topological space $Z$, a group $G$ (equipped with discrete topology) and consider the product space $Y=Z\times G$. Consider the $G$-action on $Y$ given by $$ (g, (z,h))\mapsto (z, gh). $$ Then the quotient $Y/G$ is naturally homeomorphic to $Z$, the homeomorphism $Z\to Y/G$ is given by $$ z\mapsto [(z, 1)]\in Y/G, $$ with the inverse given by $$ [(z,h)]\mapsto z. $$

Lastly:

Definition. A covering $Y\to Z$ is called trivial if for some group $G$ it is isomorphic to a covering $Z\times G\to Z$ described above.