Let $R$ be a local ring, and $S = \operatorname{Spec} R$ be its corresponding scheme. Let $s \in S$ be the closed point. Then Bosch, Lütkebohmert and Raynaud define in Néron Models
Definition The local scheme $S$ is called henselian if each étale map $X \to S$ is a local isomorphism at all points $x$ of $X$ over $s$ with trivial residue field extension $k(x) = k(s)$.
I'm not entirely sure about the quantors here: Do we only require $X \to S$ to be a local isomorphism at those points $x \mapsto s$ where $k(x) = k(s)$, or do we require $k(x) = k(s)$ for all $x \mapsto s$?
Consider a silly example: $R= k(s)=: k$.
Then for any Galois extension $L$ of $k$, you get a finite étale map $Spec(L) \to S$ which certainly isn't an isomorphism on the point of $Spec(L)$; but you really want fields to be Henselian (a Henselian ring is one where Hensel's lemma holds : solutions to equations mod $\mathfrak m$ can be lifted to integral solutions, assuming reasonable conditions).
So it must be the first interpretation (it's a local isomorphism at the points for which $k(x) = k(s)$)