I claim that according to your definition, every minimal completion is isomorphic to the Dedekind-MacNielle completion.
Suppose that $X$ is a poset. Then the Dedekind-MacNielle completion is the unique up-to isomorphism preserving $X$ complete lattice $L$ such that each $l\in L$ is the least upper bound of some subset of $X$ and the greatest lower bound of some subset of $X$.
Suppose that $X$ is a poset and $L$ is a minimal completion of $X$. Then let $M=\{\bigvee^{L}R|R\subseteq X\}$. Then $M$ is a complete lattice with partial ordering induced from $L$ with $X\subseteq M$. Therefore, by minimality, $M=L$. Similarly, let $N=\{\bigwedge^{L}R|R\subseteq L\}$. Then $N$ is a complete lattice with $X\subseteq N\subseteq L$. Again, by minimality, we conclude that $N=L$ as well. Therefore, since each $l\in L$ is the least upper bound and greatest lower bound of a subset of $L$, we conclude that $L$ is the Dedekind-MacNielle completion of $X$.