Whether or not a group action on a possibly empty set is transitive if for all elements in the set, their orbits equal the set

Question

I think the parenthetical remark in Definition 2.4 here is wrong if $X$ is empty.

If $X$ is a $G$-set and if $G · x = X$ for one (or equivalently all) $x ∈ X$, we say that $G$ acts transitively on $X$.

Why I think this is wrong:

For the first direction of one $x$ implies all $x$: The existence of an $x \in X$ implies $X$ is not empty. No problem here.

For the other direction: We could have "all $x \in X$" as holding vacuously for empty $X$. Therefore, no such $x \in X$ exists.

Do I misunderstand?