Model-theoretic grammars , also known as constraint-based grammars, contrast with generative grammars in the way they define sets of sentences: they state constraints on syntactic structure rather than providing operations for generating syntactic objects.
While “generative” is sometimes used to mean “generativist”/“transformational”/“Chomskyan”/“Minimalist”, that’s not what is meant in that article.
A model-theoretic approach to syntax views syntactic objects as fixed and states constraints on them. A generative approach gives an algorithm that enumerates them. As a simple example, consider the class of regular languages. The class can be defined generatively, using finite-state automata. Or it can be defined model-thoeretically, using monadic second-order logic over strings. The equivalence of these two definitions is the Büchi-Elgot-Trakhtenbrot theorem - Wikipedia