Waiting times between two consecutive infection and recovery events in spreading processes are often assumed to be exponentially distributed, which results in Markovian (i.e., memoryless) continuous spreading dynamics. However, this is not taking into account memory (correlation) effects and discrete interactions that have been identified as relevant in social, transportation, and disease dynamics. We introduce a framework to model continuous, discrete, and hybrid forms of (non-)Markovian susceptible-infected-recovered (SIR) stochastic processes on networks. The hybrid SIR processes that we study in this paper describe infections as discrete-time Markovian and recovery events as continuous-time non-Markovian processes, which mimic the distribution of cell cycles. Our results suggest that the effective-infection-rate description of epidemic processes fails to uniquely capture the behavior of such hybrid and also general non-Markovian disease dynamics. Providing a unifying description of general Markovian and non-Markovian disease outbreaks, we instead show that the mean transmissibility produces the same phase diagrams independent of the underlying inter-event-time distributions.