I am not able to find a rigorous definition of MSO_2 logic for arbitrary structures (which I can cite). MSO_2 for graphs is often used and defined, i.e. in On the Parameterised Intractability of Monadic Second-Order Logic - Stephan Kreutzer. The only "definition" for MSO2 on structures I found was in A logical approach to multicut problems - Gottlob, Lee.
"The definition of MSO2 can be easily generalized to arbitrary structures by allowing quantifi- cation over subsets of the input relation (notice that in the case of a graph, the input relation is just the edge re- lation). Thus, for example, if a relational symbol R is part of the signature, then a subformula (∃X ⊆ R)φ(X), expressing that there exists a subset X of the relation R such that φ(X) holds for some formula φ, could be part of an MSO2 formula."
While this is enough explanation to know what MSO2 on arbitrary structures is, I really would appreciate if someone could give me a pointer to a formal, rigorous definition somewhere in the literature.