= Solution
The relation $A\subseteq B$ means that the domain of $A$ is a <subset> of the domain of $B$ and the inclusion map is an $L$-embedding. Thus it contains all named constants, is closed under every named function, and uses the restrictions of the ambient relations.
The stronger relation $A\preccurlyeq B$ means that for every <first-order formula> $\phi(\bar x)$ and tuple $\bar a$ from $A$,
$$
\boxed{A\models\phi(\bar a)\quad\Longleftrightarrow\quad B\models\phi(\bar a).}
$$
This includes formulas with <quantifiers> ranging over the respective domains. Elementary inclusion implies ordinary <substructure of a first-order structure>[substructure] inclusion, but the latter need only preserve quantifier-free truth. In particular an existential witness available in $B$ need not lie in an arbitrary <substructure of a first-order structure>[substructure].
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