In this paper, we investigate foundational ques- tions related to the simplification based analysis of input-output stability of hybrid systems. Simulations and bisimulations are canonical notions that provide the relation between the original system and its abstraction that preserve several properties, including safety. However, recent investigations have shown that stability properties are not preserved by these notions. Hence, there have been efforts to strengthen these notions with additional requirements such as continuity on the relations, that enforce preservation of several stability notions including Lyapunov, asymptotic and input-to-state stability. Here we continue this line of work, and propose strengthenings of simu- lation/bisimulation relations for preservation of two variants of input-output stability for hybrid systems that are inspired by the incremental input-output stability and incremental state- independent input-output stability for continuous dynamical systems.