We present rigorous and intuitive master equation models to study on-demand single photon sources from pulse-excited quantum dots coupled to cavities. We consider three methods of source excitation: resonant pi-pulse, off-resonant phonon-assisted inversion, and two-photon excitation of a biexciton-exciton cascade, and investigate the effect of the pulse excitation process on the quantum indistinguishability, efficiency, and purity of emitted photons. By explicitly modelling the time-dependent pulsed excitation process in a manner which captures non-Markovian effects associated with coupling to photon and phonon reservoirs, we find that photons of near-unity indistinguishability can be emitted with over 90% efficiency for all these schemes, with the off-resonant schemes not necessarily requiring polarization filtering due to the frequency separation of the excitation pulse, and allowing for very high single photon purities. Furthermore, the off-resonant methods are shown to be robust over certain parameter regimes, with less stringent requirements on the excitation pulse duration in particular. We also derive a semi-analytical simplification of our master equation for the off-resonant drive, which gives insight into the important role that exciton-phonon decoupling for a strong drive plays in the off-resonant phonon-assisted inversion process
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