Intuition for geometric definition of Henselian (local) scheme? An excerpt from section 2.3 of Néron Models:

Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $k$. Let $S$ be the affine (local) scheme of $R$, and let $s$ be the closed point of $S$. From a geometric point of view, Henselian and strictly Henselian rings can be introduced via schemes which satisfy certain aspects of the inverse function theorem.
Definition 1. The local scheme $S$ is called Henselian if each étale map $X\to S$ is a local isomorphism at all points of $X$ over $s$ with trivial residue field extension $k(x)=k(s)$. If, in addition, the residue field $k(s)$ is separably closed, $S$ is called strictly Henselian.

What's the geometric intuition here? In other words, why do we only ask the inverse function theorem to hold at points of the special fiber with trivial residue field extension?
 A: The notion of Henselian ring (and Henselization) is rather subtle, and has appeared only in the fifties with the Japanese school (Azumaya, Nagata). It is distinct from the notion of Henselian ring introduced in the thirties by Hensel, which needs a topology and, for instance is well adapated to complete Noetherian local rings. Later, in the sixties, Grothendieck has developed these notions in full generality for the need of étale topology.
Its interest is rather specific to algebraic geometry, because in other geometries (differential geometry, complex analytic geometry) the local rings of varieties are automatically Henselian, meaning the inverse function theorem (or as I like to call it, the "local inverse theorem") is valid.
The main point is that, if we start with an étale map $f: X \to S$ of (affine) schemes, viewed as a good algebraic substitute to the notion of a local isomorphism in the complex analytic setting, then $f$ is not necessarily ${\color{red} {\text{finite}}}$. It's just quasi-finite (the fibers are finite). The delicate Main Theorem of Zariski, says $X$ can be realized as an open subscheme of some $Y$ finite over $S$. But in general, one cannot expect $Y$ to be both ${\color{red} {\text{finite}}}$ and ${\color{red} {\text{étale}}}$ over $S$. So, for this precise reason, I do not encourage you to think of Henselization in terms of fundamental groups, which deal with finite étale morphisms. My apologies to Alex Youcis.
If $s$ is a point of $S$ and $x$ is a point of $X$ above $s$ with trivial residue extension, the defect of $(S, s)$ to be Henselian reflects the difference between the local ring of $S$ at $s$ and the local ring of $X$ at $x$. As étaleness over $S$ is preserved under fiber products, local maps of this kind build in a natural way a direct filtering system and the direct limit is, so to speak, tautologically a ${\color{red} {\text{Hensenlization}}}$ of $(S, s)$ which satisfies the inverse theorem for étale local maps with trivial residual extensions.
In general, there is no other convenient description of the Henselization, making the notion rather abstract. Nevertheless, if we start with a local ${\color{red} {\text{normal}}}$ ${\color{red}{\text{excellent}}}$ Noetherian ring $A$, with maximal ideal $\mathfrak{m}$, and if we denote by $A\text{^}$ the completion of $(A, \mathfrak{m})$ and by $\mathfrak{m}\text{^}$ its maximal ideal, then $A\text{^}$ is still normal and we can look at the integral closure $B$ of $A$ in the completion $A\text{^}$ of $A$. Moreover if we localize $B$ at the pull-back of $\mathfrak{m}\text{^}$ to get a local ring $C$, one can show $C$ is in fact a Henselization of $A$. It has been the starting point of Nagata's approach.
To get the strict Henselization is just a matter of arithmetic where we increase the residue field till its separable closure. The strict Henselization $A(sh))$ of $A$ is integral and ind finite étale over the Henselization $A(h)$. Thus, this last stage $A(h) \to A(sh)$ looks as an ind étale cover but the map $A \to A(h)$ is strongly different in nature.

Hey Simone, no need to apologize--I'm grateful for your nice answer! That said, I think, perhaps, there was some confusion as to the point of my comment. I was not saying, or at least didn't intend to say, that being Henselian was equivalent, even intuitively, to the statement on fundamental groups that I made. Rather, I was thinking of this statement about fundamental groups to be symptomatic of what makes Henselian rings Henselian. A slightly better way of putting it might be that the points of the (small) étale topos are the spectra of strictly Henselian rings. Thus, we think of strictly Henselian rings as being like the 'topologically trivial' objects when thinking in terms of the étale topology. So, since Henselian rings are (again, roughly) like strictly Henselian rings with topological obstruction added only at the closed point. This causes me, again perhaps incorrectly, to intuit Henselian local rings as being 'topologically trivial neighborhoods of points'. I'd be happy to hear if you think I've made some mistake in my thinking! If so, I'd be grateful for any intuition you might have beyond the construction of the Henselization.
Oops, can't change my last comment now, but when I say 'topologically trivial neighborhoods of points' I mean 'points' in the sense of the closed point $(\text{Spec}(A/\mathfrak{m}))$ not points in the topos-theoretic sense.

To think of strict Henselization as the natural localization in the context of étale topology is certainly the good and modern point of view and was the motivation of Grothendieck. The Henselization is an intermediate localization which reduces the local study of étale maps to ${\color{red}{\text{finite}}}$ étale ones.
