Note that in (2) the morphism $x : \Spec(K) \to X$ defines a $K$-rational point of $X_z$, hence the statement makes sense. Moreover, the condition in (2) is independent of the choice of $\Spec(K) \to X$ in the equivalence class of $x$ (details omitted; this will also follow from the arguments below because the other conditions do not depend on this choice). Also note that we can always choose a diagram as in (3) by: first choosing a scheme $W$ and a surjective \'etale morphism $W \to Z$, then choosing a scheme $V$ and a surjective \'etale morphism $V \to W \times_Z Y$, and finally choosing a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Having made these choices we set $U \to W$ equal to the composition $U \to V \to W$ and we can pick a point $u \in U$ mapping to $x$ because the morphism $U \to X$ is surjective. \medskip\noindent Suppose given both a diagram as in (3) and a geometric point $\overline{x} : \Spec(k) \to X$ as in (1). By Properties of Spaces, Lemma \ref{spaces-properties-lemma-geometric-lift-to-usual} we can choose a geometric point $\overline{u} : \Spec(k) \to U$ lying over $u$ such that $\overline{x} = a \circ \overline{u}$. Denote $\overline{v} : \Spec(k) \to V$ and $\overline{w} : \Spec(k) \to W$ the induced geometric points of $V$ and $W$. In this setting we know that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ and similarly for $Y$ and $Z$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring}. In the same vein we have $$ \mathcal{F}_{\overline{x}} = (a^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh} $$ see Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-quasi-coherent}. Note that the stalk of $\mathcal{F}_w$ at $u$ is given by $$ (\mathcal{F}_w)_u = (a^*\mathcal{F})_u/\mathfrak m_w(a^*\mathcal{F})_u $$ and the local ring of $V_w$ at $v$ is given by $$ \mathcal{O}_{V_w, v} = \mathcal{O}_{V, v}/\mathfrak m_w\mathcal{O}_{V, v}. $$ Since $\mathfrak m_{\overline{z}} = \mathfrak m_w \mathcal{O}_{Z, \overline{z}} = \mathfrak m_w \mathcal{O}_{W, w}^{sh}$ we see that \begin{align*} \mathcal{F}_{\overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{F}_{\overline{x}} & = (a^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{X, \overline{x}} \\ & = (\mathcal{F}_w)_u \otimes_{\mathcal{O}_{U_w, u}} \mathcal{O}_{U, u}^{sh}/\mathfrak m_w\mathcal{O}_{U, u}^{sh} \\ & = (\mathcal{F}_w)_u \otimes_{\mathcal{O}_{U_w, u}} \mathcal{O}_{U_w, \overline{u}}^{sh} \\ & = (\mathcal{F}_w)_{\overline{u}} \end{align*} the penultimate equality by Algebra, Lemma \ref{algebra-lemma-quotient-strict-henselization} and the last equality by Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-quasi-coherent}. The same arguments applied to the structure sheaves of $V$ and $Y$ show that $$ \mathcal{O}_{V_w, \overline{v}}^{sh} = \mathcal{O}_{V, v}^{sh}/\mathfrak m_w \mathcal{O}_{V, v}^{sh} = \mathcal{O}_{Y, \overline{y}}/ \mathfrak m_{\overline{z}}\mathcal{O}_{Y, \overline{y}}. $$ OK, and now we can use Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-at-point} to see that (1) is equivalent to (3). \medskip\noindent Finally we prove the equivalence of (2) and (3). To do this we pick a field extension $\tilde K$ of $K$ and a morphism $\tilde x : \Spec(\tilde K) \to U$ which lies over $u$ (this is possible because $u \times_{X, x} \Spec(K)$ is a nonempty scheme). Set $\tilde z : \Spec(\tilde K) \to U \to W$ be the composition. We obtain a commutative diagram $$ \xymatrix{ & & & U_w \times_w \tilde z \ar[llld]_a \ar[rr] \ar[dr] & & V_w \times_w \tilde z \ar[llld]_>>>>>>>b \ar[dl] \\ X_z \ar[rr]_f \ar[dr]_g & & Y_z \ar[dl]^h &  & \tilde z \ar[llld]_c \\ & z } $$ where $z = \Spec(K)$ and $w = \Spec(\kappa(w))$. Now it is clear that $\mathcal{F}_w$ and $\mathcal{F}_z$ pull back to the same module on $U_w \times_w \tilde z$. This leads to a commutative diagram $$ \xymatrix{ X_z \ar[d] & U_w \times_w \tilde z \ar[l] \ar[d] \ar[r] & U_w \ar[d] \\ Y_z & V_w \times_w \tilde z \ar[l] \ar[r] & V_w } $$ both of whose squares are cartesian and whose bottom horizontal arrows are flat: the lower left horizontal arrow is the composition of the morphism $Y \times_Z \tilde z \to Y \times_Z z = Y_z$ (base change of a flat morphism), the \'etale morphism $V \times_Z \tilde z \to Y \times_Z \tilde z$, and the \'etale morphism $V \times_W \tilde z \to V \times_Z \tilde z$. Thus it follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-module-flat} that $$ \mathcal{F}_z\text{ flat at }x\text{ over }Y_z \Leftrightarrow \mathcal{F}|_{U_w \times_w \tilde z} \text{ flat at }\tilde x\text{ over }V_w \times_w \tilde z \Leftrightarrow \mathcal{F}_w\text{ flat at }u\text{ over }V_w $$ and we win.