(All of the morphisms occurring in this paragraph are representable by algebraic spaces, hence the conventions and results of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms} are applicable.) Assume $x$ is a finite type point. Choose an affine scheme $U$, a closed point $u \in U$, and a smooth morphism $\varphi : U \to \mathcal{X}$ with $\varphi(u) = x$, see Lemma \ref{lemma-identify-finite-type-points}. Set $u = \Spec(\kappa(u))$ as usual. Set $R = u \times_\mathcal{X} u$ so that we obtain a groupoid in algebraic spaces $(u, R, s, t, c)$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}. The projection morphisms $R \to u$ are the compositions $$ R = u \times_\mathcal{X} u \to u \times_\mathcal{X} U \to u \times_\mathcal{X} X = u $$ where the first arrow is of finite type (a base change of the closed immersion of schemes $u \to U$) and the second arrow is smooth (a base change of the smooth morphism $U \to \mathcal{X}$). Hence $s, t : R \to u$ are locally of finite type (as compositions, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-type}). Since $u$ is the spectrum of a field, it follows that $s, t$ are flat and locally of finite presentation (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation}). We see that $\mathcal{Z} = [u/R]$ is an algebraic stack by Criteria for Representability, Theorem \ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}. By Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack} we obtain a canonical morphism $$ f : \mathcal{Z} \longrightarrow \mathcal{X} $$ which is fully faithful. Hence this morphism is representable by algebraic spaces, see Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces} and a monomorphism, see Properties of Stacks, Lemma \ref{stacks-properties-lemma-monomorphism}. It follows that the residual gerbe $\mathcal{Z}_x \subset \mathcal{X}$ of $\mathcal{X}$ at $x$ exists and that $f$ factors through an equivalence $\mathcal{Z} \to \mathcal{Z}_x$, see Properties of Stacks, Lemma \ref{stacks-properties-lemma-residual-gerbe-unique}. By construction the diagram $$ \xymatrix{ u \ar[d] \ar[r] & U \ar[d] \\ \mathcal{Z} \ar[r]^f & \mathcal{X} } $$ is commutative. By Criteria for Representability, Lemma \ref{criteria-lemma-flat-quotient-flat-presentation} the left vertical arrow is surjective, flat, and locally of finite presentation. Consider $$ \xymatrix{ u \times_\mathcal{X} U \ar[d] \ar[r] & \mathcal{Z} \times_\mathcal{X} U \ar[r] \ar[d] & U \ar[d] \\ u \ar[r] & \mathcal{Z} \ar[r]^f & \mathcal{X} } $$ As $u \to \mathcal{X}$ is locally of finite type, we see that the base change $u \times_\mathcal{X} U \to U$ is locally of finite type. Moreover, $u \times_\mathcal{X} U \to \mathcal{Z} \times_\mathcal{X} U$ is surjective, flat, and locally of finite presentation as a base change of $u \to \mathcal{Z}$. Thus $\{u \times_\mathcal{X} U \to \mathcal{Z} \times_\mathcal{X} U\}$ is an fppf covering of algebraic spaces, and we conclude that $\mathcal{Z} \times_\mathcal{X} U \to U$ is locally of finite type by Descent on Spaces, Lemma \ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}. By definition this means that $f$ is locally of finite type (because the vertical arrow $\mathcal{Z} \times_\mathcal{X} U \to \mathcal{Z}$ is smooth as a base change of $U \to \mathcal{X}$ and surjective as $\mathcal{Z}$ has only one point). Since $\mathcal{Z} = \mathcal{Z}_x$ we see that (3) holds. \medskip\noindent It is clear that (3) implies (2). If (2) holds then $x$ is a finite type point of $\mathcal{X}$ by Lemma \ref{lemma-finite-type-points-morphism} and Lemma \ref{lemma-enough-finite-type-points} to see that $\mathcal{Z}_{\text{ft-pts}}$ is nonempty, i.e., the unique point of $\mathcal{Z}$ is a finite type point of $\mathcal{Z}$.