I've never seen the usage you mention before. Certainly if "theory" is said in a room full of mathematical logicians, unless someone says otherwise out loud everyone will assume it means either "set of sentences" or "deductively closed set of sentences." Occasionally specific contexts impose additional constraints - e.g. in model theory we often mean "complete consistent set of theories," and in computability theory we often mean "countable theory" - but $(i)$ those are understood as minor abuses of terminology and $(ii)$ they're still in the usual context of "theories = sets of sentences."
It's worth noting however - and this answers your first question - that every theory in the usual context corresponds to a theory of the type you describe: given a set of sentences $T$, we can look at the pair $(\Sigma,I)$ where $\Sigma$ is the language of $T$ and $I$ is the set of models of $T$. (This is mentioned in footnote $5$, which also points out that the usage here is nonstandard.) This makes that definition the (or at least a) right notion for "semantic" abstract model theory, and However, none of this contradicts the fact that this is definitely a highly nonstandard usage, and in any context where this meaning is used this will be stated explicitly.
As to where it comes from, the notion (as opposed to its name) is a natural one anytime logics other than first-order are considered; I imagine it either emerged in abstract model theory or in computer science (in the study of satisfiability over finite structures, where trans-first-order logics are actually quite important and "safe to use").