In this work, we give an overview of different notions of stability, dissipativity and passivity. In particular, we first focus our attention to incremental stability, convergence dynamics, and contractive systems. We then consider their extension to the case of open systems, i.e., dynamical systems that interact with the surrounding environment via inputs and output signals, hence we address different concepts of dissipativity/passivity, such as incremental, convergence and differential dissipativity/passivity. We write the notions into an unifying framework in order to make an easier comparison among them by leveraging coordinate-free definitions. We then obtain conditions to determine the related storage functions for a class of passive system, i.e., port-Hamiltonian dynamics. In particular, we show what additional conditions must be satisfied by a passive system with its associated storage function in order to fit the non-standard passivity definitions. We obtain the relative storage functions based on the knowledge of the system Hamiltonian that for these systems plays the role of storage function for the standard passivity notion.