= Solution
Use the local-presentation definition of a <quasi-coherent sheaf>: locally it admits a presentation $\mathcal O^{(I)}\to\mathcal O^{(J)}\to\mathcal F\to0$ by direct sums of copies of the <structure sheaf>, with no finiteness requirement on the index sets. If $M$ is an $A$-<module>, choose a free <module> presentation $A^{(I)}\to A^{(J)}\to M\to0$. Exactness of <localization of a module> gives the corresponding local presentation of $\widetilde M$. Hence a <sheaf> which is associated with a <module> on every <affine open subset> is a <quasi-coherent sheaf>.
For the converse, begin with a <quasi-coherent sheaf> and a local presentation on an <affine open subset> $V=\operatorname{Spec}R$. The free <sheaves> have <global sections> $R^{(I)}$ and $R^{(J)}$: a section has locally finite support, and quasi-compactness of $V$ makes its support finite globally. Part (i) identifies the presentation map with an $R$-<module homomorphism>. Exactness of <localization> then identifies its <cokernel sheaf> with $\widetilde M_V$ for the <module> cokernel $M_V$. Thus there is an <affine open cover> on which $\mathcal F$ is associated with <modules>.
It remains to show this on an arbitrary <affine open subset> $U=\operatorname{Spec}A$, rather than only the selected cover. Around each point of $U$, choose an affine neighborhood contained in $U$ with such a local presentation. Choose a finite cover $U=\bigcup_{i=1}^rD(s_i)$ refining these neighborhoods. On each $D(s_i)$ the restriction is associated with a <module>. Its restriction to $D(s_is_j)$ is therefore the corresponding <localization>, and the same is true after intersecting with $D(a)$. Put $M=\Gamma(U,\mathcal F)$. The <sheaf gluing axiom> gives
$$
M=\ker\left(\prod_i\Gamma(D(s_i),\mathcal F)\longrightarrow\prod_{i,j}\Gamma(D(s_is_j),\mathcal F)\right),
$$
where the arrow is the difference of the two restriction maps. <Localization of a module> commutes with kernels and finite products. Localizing this formula at $a$ therefore gives exactly the gluing formula for $D(a)$, covered by the $D(as_i)$. Consequently
$$
M_a\simeq\Gamma(D(a),\mathcal F),
$$
compatibly with restriction. These isomorphisms on the <principal open subsets> identify $\widetilde M$ with $\mathcal F|_U$. We have proved the <global-section localization for quasi-coherent sheaves> and, in particular,
$$
\boxed{\mathcal F|_U\simeq\widetilde{\Gamma(U,\mathcal F)}\quad\text{for every affine open }U.}
$$
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