The dynamics of open quantum systems is often solved by stochastic unravelings where the average over the state-vector realizations reproduces the density matrix evolution. We focus on quantum-jump descriptions based on the rate-operator formalism. In addition to displaying and exploiting different equivalent ways of writing the master equation, we introduce state-dependent rate-operator transformations within the framework of stochastic pure state realizations, allowing us to extend and generalize the previously developed formalism. As a consequence, this improves the controllability of the stochastic realizations and subsequently greatly benefits when searching for optimal simulation schemes to solve open system dynamics. At a fundamental level, intriguingly, our results show that it is possible to have positive unravelings, without reverse quantum jumps and avoiding the use of auxiliary degrees freedom, in a number of example cases even when the corresponding dynamical map breaks the property of P divisibility, thus being in the strongly non-Markovian regime.