<a
href="https://en.wikipedia.org/wiki/Antimatroid">Antimatroids</a>
are a good example. We have the syllogism "Antimatroids are to
matroids as join-distributive lattices are to geometric lattices."
Two other examples are the characterizations of complemented modular
lattices of finite length $n\geq 4$ and primary modular lattices of
finite length $n\geq 4$. (A modular lattice of finite length is
*primary* if for every join-irreducible $t$, the interval
$[\hat{0},t]$ is a chain, and dually, e.g., the lattice of subgroups of
a finite abelian group.) Their characterizations are analogous to
characterizing finite distributive lattices as a collection of sets
closed under union and intersection. See for instance Theorems 5 and 6
of Alan Day, Geometric applications in modular lattices, in
*Universal Algebra and Lattice Theory (Puebla, 1982)*, Springer
Lecture Notes in Mathematics **1004**, pp. 111-141. I don't know a
common generalization of distributive and geometric lattices with the
type of structure asked for.