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.)