Yes that is an instance of the connection between algebraic notions of equivalence in process algebra and the temporal logical equivalence. Adequacy of logic [HM] with respect to a given equivalence relation on language of processes is one notion that has been use to relate the notions back to logic. I am hoping for reference where these notions of equivalence are systematically studied as relation between two structures with respect to the first order logic. These structures can then be "later" thought of as finite state automata or transition systems.