Assume (1), i.e., let $\varphi : U \to X$ be an \'etale morphism from a scheme towards $X$, and let $u, u'$ be points of $U$ lying over $x$ such that the fibre of $|R| \to |U| \times_{|X|} |U|$ over $(u, u')$ is a finite set. In this proof we think of a point $u = \Spec(\kappa(u))$ as a scheme. Note that $u \to U$, $u' \to U$ are monomorphisms (see Schemes, Lemma \ref{schemes-lemma-injective-points-surjective-stalks}), hence $u \times_X u' \to R = U \times_X U$ is a monomorphism. In this language the assumption really means that $u \times_X u'$ is a scheme whose underlying topological space has finitely many points. Let $\psi : W \to X$ be an \'etale morphism from a scheme towards $X$. Let $w, w' \in W$ be points of $W$ mapping to $x$. We have to show that $w \times_X w'$ is a scheme whose underlying topological space has finitely many points. Consider the fibre product diagram $$ \xymatrix{ W \times_X U \ar[r]_p \ar[d]_q & U \ar[d]^\varphi \\ W \ar[r]^\psi & X } $$ As $x$ is the image of $u$ and $u'$ we may pick points $\tilde w, \tilde w'$ in $W \times_X U$ with $q(\tilde w) = w$, $q(\tilde w') = w'$, $u = p(\tilde w)$ and $u' = p(\tilde w')$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. As $p$, $q$ are \'etale the field extensions $\kappa(w) \subset \kappa(\tilde w) \supset \kappa(u)$ and $\kappa(w') \subset \kappa(\tilde w') \supset \kappa(u')$ are finite separable, see Remark \ref{remark-recall}. Then we get a commutative diagram $$ \xymatrix{ w \times_X w' \ar[d] & \tilde w \times_X \tilde w' \ar[l] \ar[d] \ar[r] & u \times_X u' \ar[d] \\ w \times_X w' & \tilde w \times_S \tilde w' \ar[l] \ar[r] & u \times_S u' } $$ where the squares are fibre product squares. The lower horizontal morphisms are \'etale and quasi-compact, as any scheme of the form $\Spec(k) \times_S \Spec(k')$ is affine, and by our observations about the field extensions above. Thus we see that the top horizontal arrows are \'etale and quasi-compact and hence have finite fibres. We have seen above that $|u \times_X u'|$ is finite, so we conclude that $|w \times_X w'|$ is finite. In other words, (2) holds. \medskip\noindent Assume (2). Let $U \to X$ be an \'etale morphism from a scheme $U$ such that $x$ is in the image of $|U| \to |X|$. Let $u \in U$ be a point mapping to $x$. Then we have seen in the previous paragraph that $u = \Spec(\kappa(u)) \to X$ has the property that $u \times_X u$ has a finite underlying topological space. On the other hand, the projection maps $u \times_X u \to u$ are the composition $$ u \times_X u \longrightarrow u \times_X U \longrightarrow u \times_X X = u, $$ i.e., the composition of a monomorphism (the base change of the monomorphism $u \to U$) by an \'etale morphism (the base change of the \'etale morphism $U \to X$). Hence $u \times_X U$ is a disjoint union of spectra of fields finite separable over $\kappa(u)$ (see Remark \ref{remark-recall}). Since $u \times_X u$ is finite the image of it in $u \times_X U$ is a finite disjoint union of spectra of fields finite separable over $\kappa(u)$. By Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field} we conclude that $u \times_X u$ is a finite disjoint union of spectra of fields finite separable over $\kappa(u)$. In other words, we see that $u \times_X u \to u$ is quasi-compact and \'etale. This means that (3) holds. \medskip\noindent Let us prove that (3) implies (4). Let $\Spec(k) \to X$ be a morphism from the spectrum of a field into $X$, in the equivalence class of $x$ such that the two projections $t, s : R = \Spec(k) \times_X \Spec(k)  \to \Spec(k)$ are quasi-compact and \'etale. This means in particular that $R$ is an \'etale equivalence relation on $\Spec(k)$. By Spaces, Theorem \ref{spaces-theorem-presentation} we know that the quotient sheaf $X' = \Spec(k)/R$ is an algebraic space. By Groupoids, Lemma \ref{groupoids-lemma-quotient-groupoid-restrict} the map $X' \to X$ is a monomorphism. Since $s, t$ are quasi-compact, we see that $R$ is quasi-compact and hence Properties of Spaces, Lemma \ref{spaces-properties-lemma-point-like-spaces} applies to $X'$, and we see that $X' = \Spec(k')$ for some field $k'$. Hence we get a factorization $$ \Spec(k) \longrightarrow \Spec(k') \longrightarrow X $$ which shows that $\Spec(k') \to X$ is a monomorphism mapping to $x \in |X|$. In other words (4) holds. \medskip\noindent Finally, we prove that (4) implies (1). Let $\Spec(k) \to X$ be a monomorphism with $k$ a field in the equivalence class of $x$. Let $U \to X$ be a surjective \'etale morphism from a scheme $U$ to $X$. Let $u \in U$ be a point over $x$. Since $\Spec(k) \times_X u$ is nonempty, and since $\Spec(k) \times_X u \to u$ is a monomorphism we conclude that $\Spec(k) \times_X u = u$ (see Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field}). Hence $u \to U \to X$ factors through $\Spec(k) \to X$, here is a picture $$ \xymatrix{ u \ar[r] \ar[d] & U \ar[d] \\ \Spec(k) \ar[r] & X } $$ Since the right vertical arrow is \'etale this implies that $k \subset \kappa(u)$ is a finite separable extension. Hence we conclude that $$ u \times_X u = u \times_{\Spec(k)} u $$ is a finite scheme, and we win by the discussion of the meaning of property (1) in the first paragraph of this proof.