In the affine case this is Algebra, Lemma \ref{algebra-lemma-finite-flat-module-finitely-presented}. The scheme case does not follow directly from the affine case, so we simply repeat the arguments. \medskip\noindent Assume (1). Consider a closed immersion $i : Z \to X$ such that $i$ is flat. Then $i_*\mathcal{O}_Z$ is quasi-coherent and flat, hence finite locally free by (1). Thus $Z = \text{Supp}(i_*\mathcal{O}_Z)$ is also open and we see that (2) holds. Hence the implication (1) $\Rightarrow$ (2) follows from the characterization of flat closed immersions in Lemma \ref{lemma-characterize-flat-closed-immersions}. \medskip\noindent For the converse assume that $X$ satisfies (2). Let $\mathcal{F}$ be a finite flat quasi-coherent $\mathcal{O}_X$-module. The support $Z = \text{Supp}(\mathcal{F})$ of $\mathcal{F}$ is closed, see Modules, Lemma \ref{modules-lemma-support-finite-type-closed}. On the other hand, if $x \leadsto x'$ is a specialization, then by Algebra, Lemma \ref{algebra-lemma-finite-flat-local} the module $\mathcal{F}_{x'}$ is free over $\mathcal{O}_{X, x'}$, and $$ \mathcal{F}_x = \mathcal{F}_{x'} \otimes_{\mathcal{O}_{X, x'}} \mathcal{O}_{X, x}. $$ Hence $x' \in \text{Supp}(\mathcal{F}) \Rightarrow x \in \text{Supp}(\mathcal{F})$, in other words, the support is closed under generalization. As $X$ satisfies (2) we see that the support of $\mathcal{F}$ is open and closed. The modules $\wedge^i(\mathcal{F})$, $i = 1, 2, 3, \ldots$ are finite flat quasi-coherent $\mathcal{O}_X$-modules also, see Modules, Section \ref{modules-section-symmetric-exterior}. Note that $\text{Supp}(\wedge^{i + 1}(\mathcal{F})) \subset \text{Supp}(\wedge^i(\mathcal{F}))$. Thus we see that there exists a decomposition $$ X = U_0 \amalg U_1 \amalg U_2 \amalg \ldots $$ by open and closed subsets such that the support of $\wedge^i(\mathcal{F})$ is $U_i \cup U_{i + 1} \cup \ldots$ for all $i$. Let $x$ be a point of $X$, and say $x \in U_r$. Note that $\wedge^i(\mathcal{F})_x \otimes \kappa(x) = \wedge^i(\mathcal{F}_x \otimes \kappa(x))$. Hence, $x \in U_r$ implies that $\mathcal{F}_x \otimes \kappa(x)$ is a vector space of dimension $r$. By Nakayama's lemma, see Algebra, Lemma \ref{algebra-lemma-NAK} we can choose an affine open neighbourhood $U \subset U_r \subset X$ of $x$ and sections $s_1, \ldots, s_r \in \mathcal{F}(U)$ such that the induced map $$ \mathcal{O}_U^{\oplus r} \longrightarrow \mathcal{F}|_U, \quad (f_1, \ldots, f_r) \longmapsto \sum f_i s_i $$ is surjective. This means that $\wedge^r(\mathcal{F}|_U)$ is a finite flat quasi-coherent $\mathcal{O}_U$-module whose support is all of $U$. By the above it is generated by a single element, namely $s_1 \wedge \ldots \wedge s_r$. Hence $\wedge^r(\mathcal{F}|_U) \cong \mathcal{O}_U/\mathcal{I}$ for some quasi-coherent sheaf of ideals $\mathcal{I}$ such that $\mathcal{O}_U/\mathcal{I}$ is flat over $\mathcal{O}_U$ and such that $V(\mathcal{I}) = U$. It follows that $\mathcal{I} = 0$ by applying Lemma \ref{lemma-characterize-flat-closed-immersions}. Thus $s_1 \wedge \ldots \wedge s_r$ is a basis for $\wedge^r(\mathcal{F}|_U)$ and it follows that the displayed map is injective as well as surjective. This proves that $\mathcal{F}$ is finite locally free as desired.