Here are four possible definitions for an etale, finite, surjective map $X\rightarrow Y$ between integral schemes to be considered Galois:

1. There exists a finite group $G$, and an action $\varphi: G\times X\rightarrow X$, so that the induced map $\varphi\times p_2: G\times X\rightarrow X\times_Y X$ is an isomorphism.
2. There exists a finite group $G$, and an action $\varphi: G\times X\rightarrow X$, so that $G$ acts freely an transitively on the geometric fibers of $X\rightarrow Y$.
3. There exists a finite group $G$, and an action $\varphi: G\times X\rightarrow X$, so that the extension of function fields $\kappa(X)/\kappa(Y)$ is Galois with group isomorphic to $G$.
4. The extension of function fields $\kappa(X)/\kappa(Y)$ is Galois.

Obviously, 1 implies 2 implies 3 implies 4. Note that the difference between 3 and 4 is that in 4 you are not guaranteed that the action extends from the generic point to $X$. I am inclined to believe that 4 does not imply 3, but I can't think of a counterexample.

I think that definitions 1 and 2 are probably equivalent. This is because definition 2 is probably the same as saying that $\varphi\times p_2$ is a bijection on geometric points, which, at least if $X$ and $Y$ are varieties, should imply that it is an isomorphism.

I am completely in the dark about whether or not definitions 2 and 3 are equivalent. I'm not sure what I should believe...

So:

Questions

1. Is 1 equivalent to 2?
2. Is 2 equivalent 3?
3. What is a counterexample for the equivalence of 3 and 4?
4. Is it at all helpful to ask that $X$ and $Y$ be smooth for any of these definitions? Any other adjectives that need adding?