= Solution
A <first-order theory> $T$ is <model-complete> if every <structure embedding> between models of $T$ is an <elementary embedding>. Equivalently, whenever $\mathcal M\subseteq\mathcal N$ and both satisfy $T$, one has $\mathcal M\preccurlyeq\mathcal N$:
$$
\boxed{\mathcal M\models\varphi(\bar a)\iff\mathcal N\models\varphi(\bar a)}
$$
for every <first-order formula> $\varphi$ and every finite tuple $\bar a\in M$. This is preservation of all formulas with parameters, rather than merely all <first-order sentences>.
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