What general set-theoretical definitions of the notion of "structure" are there?
By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that
a set $x$ is a structure iff $\Phi(x)$
The definitions of structures in standard model theory would qualify, but they are restricted to first-order structures (or second-order, at most).
Suppes' set-theoretical predicates à la
a set $x$ is an algebraic structure iff $\phi(x)$
would qualify, but they are obviously too specific. (A rigorous meta-definition of "set-theoretical predicate" seems to be missing.)
Bourbaki's definition of species of structures seems to be the most general one and most strongly related to categories (see here and here). Nevertheless it was never widely accepted and used.
Are there other approaches I'm just not aware of? Or is such a general definition considered useless?