Let $G$ be a group. A $G$-torsor is a set $X$ together with an action of $G$ such that for all $x,y \in X$ there is exactly one $g \in G$ such that $gx=y$. This looks like a group which has forgotten its identity (no pun intended).
Usually it is assumed that $X$ is nonempty (or more generally an inhabited object according to the nlab article), but it seems to me that it makes perfect sense also to allow $X = \emptyset$ because of the equivalent definition using Heaps. This equivalence also suggests that $X = \emptyset$ should imply that $G$ is trivial. But this is not guaranteed by the definition given above.
Question: What is a natural definition of torsors which also includes the empty set with the action of the trivial group (the empty torsor)?
In the case of sets as above, we may just add that the action is faithful, i.e. the homomorphism $G \to \text{Sym}(X)$ is injective. But how can we give a definition for, say, group schemes acting on schemes, without making a nasty case distinction?
Also I would like to know if you agree with me that it is natural to include the empty torsor. It will be an initial object in the category of torsors and as I said, especially in the definition of a heap the assumption of being nonempty seems to be artificial according to universal algebra.
[added] Thank you for all the good answers. I agree that it's not natural to consider an empty torsor. A $G$-torsor should be something which is locally isomorphic to $G$.