Context. Classical Be stars have been established as pulsating stars. Space-based photometric monitoring missions contributed significantly to that result. However, whether Be stars are just rapidly rotating SPB or β Cep stars, or whether they have to be understood differently, remains debated in the view of their highly complex power spectra. Aims. Kepler data of three known Be stars are re-visited to establish their pulsational nature and assess the properties of additional, non-pulsational variations. The three program stars turned out to be one inactive Be star, one active, continuously outbursting Be star, and one Be star transiting from a non-outbursting into an outbursting phase, thus forming an excellent sample to distill properties of Be stars in the various phases of their life-cycle. Methods. The Kepler data was first cleaned from any long-term variability with Lomb-Scargle based pre-whitening. Then a LombScargle analysis of the remaining short-term variations was compared to a wavelet analysis of the cleaned data. This offers a new view on the variability, as it enables us to see the temporal evolution of the variability and phase relations between supposed beating phenomena, which are typically not visualized in a Lomb-Scargle analysis. Results. The short-term photometric variability of Be stars must be disentangled into a stellar and a circumstellar part. The stellar part is on the whole not different from what is seen in non-Be stars. However, some of the observed phenomena might be to be due to resonant mode coupling, a mechanism not typically considered for B-type stars. Short-term circumstellar variability comes in the form of either a group of relatively well-defined, short-lived frequencies during outbursts, which are called Štefl frequencies, and broad bumps in the power spectra, indicating aperiodic variability on a time scale similar to typical low-order g-mode pulsation frequencies, rather than true periodicity. Conclusions. From a stellar pulsation perspective, Be stars are rapidly rotating SPB stars, that is they pulsate in low order g-modes, even if the rapid rotation can project the observed frequencies into the traditional high-order p-mode regime above about 4 c/d. However, when a circumstellar disk is present, Be star power spectra are complicated by both cyclic, or periodic, and aperiodic circumstellar phenomena, possibly even dominating the power spectrum.