= Solution
Here <parameter-free definability> is intended: parameters are not silently added to the <first-order language>. A subset $D\subseteq A^n$ is a <definable set> when some $L$-<first-order formula> $\varphi(x_1,\ldots,x_n)$ satisfies
$$
\boxed{D=\{\mathbf a\in A^n:\mathcal A\models\varphi(\mathbf a)\}.}
$$
A <definable function> is one whose graph is a definable set. Thus $f:A^n\to A$ is definable when an $L$-formula $\theta(\mathbf x,y)$ satisfies
$$
\boxed{\mathcal A\models\theta(\mathbf a,b)\iff b=f(\mathbf a).}
$$
In particular $\mathcal A\models\forall\mathbf x\exists!y\,\theta(\mathbf x,y)$. Definability over a parameter set $B$ permits parameters from $B$ in these <first-order formulas>, equivalently new constants naming those parameters. That is a different requirement: allowing a parameter naming an element would immediately define its singleton.
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