We first prove that (1) implies (2). Assume (1) holds. The statement is local on $X$, hence we may assume $X$ is affine. Then $U$ is quasi-compact, hence $\text{Ass}(\mathcal{F})$ is finite (Divisors, Lemma \ref{divisors-lemma-finite-ass}). Thus we may argue by induction on the number of associated points. Let $x \in U$ be a generic point of an irreducible component of the support of $\mathcal{F}$. By Divisors, Lemma \ref{divisors-lemma-finite-ass} we have $x \in \text{Ass}(\mathcal{F})$. By our choice of $x$ we have $\dim(\mathcal{F}_x) = 0$ as $\mathcal{O}_{X, x}$-module. Hence $\mathcal{F}_x$ has finite length as an $\mathcal{O}_{X, x}$-module (Algebra, Lemma \ref{algebra-lemma-support-point}). Thus we may use induction on this length. \medskip\noindent Set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_x}$. This is a coherent $\mathcal{O}_X$-module by assumption. We have $\mathcal{G}_x = \kappa(x)$. Choose a nonzero map $\varphi_x : \mathcal{F}_x \to \kappa(x) = \mathcal{G}_x$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-map-stalks-local-map} there is an open $x \in V \subset U$ and a map $\varphi_V : \mathcal{F}|_V \to \mathcal{G}|_V$ whose stalk at $x$ is $\varphi_x$. Choose $f \in \Gamma(X, \mathcal{O}_X)$ which does not vanish at $x$ such that $D(f) \subset V$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} (for example) we see that $\varphi_V$ extends to $f^n\mathcal{F} \to \mathcal{G}|_U$ for some $n$. Precomposing with multiplication by $f^n$ we obtain a map $\mathcal{F} \to \mathcal{G}|_U$ whose stalk at $x$ is nonzero. Let $\mathcal{F}' \subset \mathcal{F}$ be the kernel. Note that $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$, see Divisors, Lemma \ref{divisors-lemma-ses-ass}. Since $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) - 1$ we may apply the induction hypothesis to conclude $j_*\mathcal{F}'$ is coherent. Since $\mathcal{G} = j_*(\mathcal{G}|_U) = j_*i_{x, *}\mathcal{O}_{W_x}$ is coherent, we can consider the exact sequence $$ 0 \to j_*\mathcal{F}' \to j_*\mathcal{F} \to \mathcal{G} $$ By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $j_*\mathcal{F}$ is quasi-coherent. Hence the image of $j_*\mathcal{F}$ in $j_*(\mathcal{G}|_U)$ is coherent by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. Finally, $j_*\mathcal{F}$ is coherent by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-abelian-Noetherian}. \medskip\noindent Assume (2) holds. Exactly in the same manner as above we reduce to the case $X$ affine. We pick $x \in \text{Ass}(\mathcal{F})$ and we set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_x}$. Then we choose a nonzero map $\varphi_x : \mathcal{G}_x = \kappa(x) \to \mathcal{F}_x$ which exists exactly because $x$ is an associated point of $\mathcal{F}$. Arguing exactly as above we may assume $\varphi_x$ extends to an $\mathcal{O}_U$-module map $\varphi : \mathcal{G}|_U \to \mathcal{F}$. Then $\varphi$ is injective (for example by Divisors, Lemma \ref{divisors-lemma-check-injective-on-ass}) and we find an injective map $\mathcal{G} = j_*(\mathcal{G}|_V) \to j_*\mathcal{F}$. Thus (1) holds.