Tim-

Emil answered the question you asked, but since you wrote ``what I'd really like to know is: which finite semilattices are retracts (via $\bot$, $\vee$-preserving maps) of finite powerset lattices'' let me add to his answer.

The $2$-element semilattice is injective in the class of semilattices.

The class of injectives is closed under products and retracts.

Up to isomorphism, the powers of the $2$-element semilattice are the power-set semilattices. Hence retracts of power-set semilattices must be injective.

Conversely, since the $2$-element semilattice is the only subdirectly irreducible semilattice, every semilattice is embeddable in some power $2^S$. And since an injective is a retract of any extension, it follows that every injective arises as a retract of some $2^S$,

Thus, the retracts of the power-set semilattices, $2^S$, are exactly the injective semilattices.

Theorem 2.8 of

The Category of Semilattices

ALFRED HORN and NAOKI KIMURA

Algebra universalis 1 (1971), 26-38.

proves that a (meet-)semilattice is injective iff it is complete and satisfies an infinite distributive law, namely that the meet distributes over infinite joins.

It follows that a finite semilattice is a retract of a power-set semilattice iff it is the semilattice reduct of a finite distributive lattice. (This shows that the examples in your last struckout paragraph exhaust all examples.)