I'm not sure if this is exactly the sort of thing you are looking for, but call a category a "Boolean logos" just in case it has finite limits, the subobjects of any object form a Boolean algebra under inclusion, and pullback along any morphism induces a homomorphism of such Boolean algebras with left and right adjoints. [Much of this definition is redundant, but no matter]. Boolean logoses, functors between them which preserve all the defining structure, and arbitrary natural transformations between those comprise a two-category, which I'll call BoolLog.
Up to equivalence, the categories of all models of some theory in multi-sorted classical first-order logic with equality and the elementary embeddings between them are precisely those of the form BoolLog[B, Set]. That is, a category C is equivalent to one of the form Mod*(T) just in case there is some Boolean logos B such that C is equivalent to the full subcategory of Set^B consisting of those functors which preserve Boolean logos structure.
This isn't really saying much (the definition of a Boolean logos is a very straightforward categorical rendering of the definition of multi-sorted classical first-order logic with equality [although, like I said, it could be pared down a bit]), but, as I understand the reply you gave to my comment above, perhaps such categorical rendering is all you were looking for. (And, of course, we can easily tweak the former definition around a little to correspond to variants on the latter (e.g., the single-sorted case, the intuitionistic case, etc.))