Statistical characterization of the signal-to-interference-plus-noise ratio (SINR) via its cumulative distribution function (CDF) is ubiquitous in a vast majority of technical contributions in the area of cellular networks since it boils down to averaging the Laplace transform of the aggregate interference, a benefit accorded at the expense of confinement to the simplistic Rayleigh fading. In this work, to capture diverse fading channels that appear in realistic outdoor/indoor wireless communication scenarios, we tackle the problem differently. By exploting the moment generating function (MGF) of the SINR, we succeed in analytically assessing cellular networks performance over the shadowed κ-μ, κ-μ and η-μ fading models. The latter offer high flexibility by capturing diverse fading channels including Rayleigh, Nakagami-m, Rician, and Rician shadow fading distributions. These channel models have been recently praised for their capability to accurately model dense urban environments, future femtocells and device-to-device (D2D) shadowed channels. In addition to unifying the analysis for different channel models, this work integrates, for the first time, the coverage, the achievable rate, and the bit error probability (BEP), which are largely treated separately in the literature. The developed model and analysis are validated over a broad range of simulation setups and parameters.