= Solution
An $\mathcal O_X$-module $\mathcal F$ is a <quasi-coherent sheaf> when every affine open $V=\operatorname{Spec}A\subseteq X$ has $\mathcal F|_V\cong\widetilde M$ for some $A$-module $M$. For a morphism $f:X\to Y$, its <direct image sheaf> is
$$
(f_*\mathcal F)(V)=\mathcal F(f^{-1}V),
$$
while the <pullback of a sheaf of modules> is
$$
f^*\mathcal G=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal G.
$$
If $i:Z\hookrightarrow X$ is a <closed immersion> of <Noetherian scheme>[Noetherian schemes], then on an affine open $V=\operatorname{Spec}A\subseteq X$ one has $Z\cap V=\operatorname{Spec}(A/I)$. The restriction of $i_*\mathcal O_Z$ corresponds to the cyclic $A$-module $A/I$, so it is finitely generated. Hence $i_*\mathcal O_Z$ is a <coherent sheaf>; more generally this is the <direct image of a coherent sheaf under a closed immersion>.
Coherence need not survive an arbitrary pushforward. Let
$$
j:D(x)=\operatorname{Spec}k[x,x^{-1}]\hookrightarrow\operatorname{Spec}k[x]
$$
be the <open immersion>. The sheaf $\mathcal O_{D(x)}$ is coherent, but
$$
\Gamma(\mathbb A^1,j_*\mathcal O_{D(x)})=k[x,x^{-1}]
$$
is not a finitely generated $k[x]$-module. Therefore $j_*\mathcal O_{D(x)}$ is not coherent.
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