[Really the natural map is a poset embedding (also the inverse from the image of $P$ to $P$ is isotone). This can be seen along the preceding lines, or one can see Gratzer's book where one finds much more general results about free embeddings of partial lattices in lattices.]
Taking as $P$ the anti-chain with $n>2$ elements one sees that $L(P)$ is the free object (in the quasivariety) on $n$-generators; in particoularparticular it is generally not the Dedekind-McNeillMcNeille completion of $P$ (which is the projective line with $n$ points, hence $2$-distributive, a quite strong lattice identity). If you look at the directions of the arrows for the universal property of the Dedekind-McNeille completion (among all completions) and for the above $L(P)$ you will find this non-coincidence unsurprising (and you will find another example where lattices have too many kinds of useful morphisms to look at them with only one category).
[The last paragraph is not quite correct. I wanted to express the fact that the relation between the cut completion and the freely generated lattice resembles the relation between the one-point (minimal) compactificication and the universal (maximal) Stone compactification. However I partly messed up the reasons for this (it is a question of different arrows, but not of direction). A better analysis follows]
we have to construct a semilattice $\mathcal{M}$ together with an order embedding $i$ of $P$ in $\mathcal{M}$, such that each order-embedding $j_0$ from $P$ to a semilattice $L$ can be "canonically" extended into an order-embedding $j$ from $\mathcal{M}$ to $L$ with $j_0=j\circ i$
Thank you for the pointer to your extra-question formalization of "minimal".
For a poset $P$, you essentially consider the class of all posets $Q$ that have $P$ as sub-poset, and the following preorder: $Q\preceq Q'$ iff there is a poset embedding of $Q$ in $Q'$ that fixes each element of $P$.
I see no reasons for $Q\preceq Q'\preceq Q$ to imply that $Q,Q'$ are isomorphic extensions of $P$ (i.e. I see no reasons for the composition of the two embeddings to be the identity also outside $P$).
When considering the subclass of join-semilattices (or lattices, or meet-semilattices; also complete lattices can be considered) $Q$, you are searching minimum (not only minimal!) elements for the preorder on the subclass; lacking antisymmetry, there might be non-isomorphic minima.
{\em{When $P$ is finite then the Dedekind - McNeille completion is minimal: as lattice, as join semilattice, as meet semilattice.}}
{\em{When $P$ is generic then the Dedekind - McNeille completion is minimal as complete lattice.}}
The elements of such completion (Dedekind cuts) are pairs $(A,B)$ of subsets of $P$ such that: (1) $A$ is the set of elements $a$ of $P$ such that $a\leq b$ for each $b$ in $B$; (2) $B$ is the set of elements $b$ of $P$ such that $a\leq b$ for each $a$ in $A$.
The ordering on such pairs is inclusion between the first components, or equivalently dual inclusion between the second components.
Equivalently: the binary relation $\leq$ between $P$ and $P$ has an associated Galois connection; the $A$ are the closed sets in the first component, and the $B$ the corresponding closed sets in the second component.
In any join-semilattice $Q$ extending $P$, the elements Sup$A$ exist in $Q$ since $A$ is finite; dually, Inf$B$ exist in meet-semilattice extensions of $P$.
Using the map ``from $A$ to Sup$A$'',one order-embeds the Dedekind - McNeille completion of $P$ in any join-semilattice extension $Q$ of $P$:
the map ``fixes'' $P$ (if $A$ is the interval $(p]$ ending in $p$ then Sup$A$ is $p$); the map is isotone (if $A$ is contained in $A'$ then Sup$A$ is contained in Sup$A'$); conversely, if Sup$A$ is contained in Sup$A'$ then the interval [Sup$A$) (i.e. the set $B_Q$ of elements in $Q$ which contain each $a$ in $A$) contains $B'_Q$; hence $B=B_Q\cap P$ contains $B'=B'_Q\cap P$, hence $A$ is contained in $A'$.
The same argument also gives two order embeddings of the Dedekind - McNeille completion of a possibly infinite $P$ in any complete (semi)lattice $Q$ extending $P$. These two embeddings coincide iff each element of $Q$ is both sup and inf of (different) subsets of $P$, i.e. iff $Q$ is the Dedekind - McNeille completion of $P$. You can easily see the two embeddings in the specific example embedded in your question.
[Joseph Van Name has already given elsewhere very similar arguments]
{\em{This minimum is unique up to isomorphisms (as poset extension of the finite poset $P$).}}
For a finite $P$, let $Q$ a minimum poset extension of $P$ among semilattices; this minimum conditions implies that $Q$ is poset embedded, respecting $P$, in the Dedekind - McNeille completion of $P$.
Since $P$ is finite, also its Dedekind - McNeille completion is finite, hence also is the subposet $Q$.
One has two finite sets ($Q$ and the Dedekind - McNeille completion of $P$) each with a injective map in the other; they must have the same cardinality and the poset embeddings of one in the other must be surjective, hence isomorphisms.
For a infinite $P$, even when each cut $(A,B)$ is finitely generated ($B$ is the polar of a finite subset of $A$ and $A$ is the polar of a finite subset of $B$), the preceding argument fails since inside the cut completion of $Q$ one can have Sup$A$ strictly less than Inf$B$, so Sup of finite subsets of $A$ containing the generators need not coincide. In fact one has:
{\em{when $P$ is a countably infinite antichain, then $P$ has no minimal lattice or semilattice extension}}
Consider as $P$ the antichain $a_0,a_1,a_2,\dots$ and the join semilattice $Q$ obtained by adding a chain of distinct elements $b_n$ as sup of $a_0,a_1,\dots,a_n$. [One can draw a picture with the $a$ on a horizontal line and the $b$ on a vertical line].
This already shows that the obvious candidate (the join semilattice $Q'$ generated by $P$ in its cut completion, i.e. the antichain with a maximum element added) is not minimal since it cannot be embedded in $Q$ (in $Q$ no element contains infinitely many elements of $P$).
Conversely, $Q$ is not embeddable in $Q'$ since $Q$ has more than one element outside $P$ and $Q'$ has only one. Really, this shows that no embedding in $Q'$ exists for a poset with at least two elements outside $P$, and since $Q'$ is the only join semilattice with only one element outside $P$ the thesis follows.
[One could consider a modified definition of minimal in a given class: a extension of $P$ in the class such that no proper sub-extension is again in the class. The cut completion, in the above cases where it is the absolute minimum, is also the unique minimal in such cases. I see no reasons for a arbitrary extension in one of the three wanted classes to contain a sub-extension which is minimal: the above $Q$ is a join-semilattice extending the antichian $P$, but one can leave only a infinite subsequence of the $b_i$ and again have a join-semilattice extension (but not a sub-semilattice of $Q$), so $Q$ contains no minimal extension. On the other hand $Q'$ is clearly minimal in this modified sense]