This paper shows that the recently proposed $κ$-$μ$ shadowed fading model includes, besides the $κ$-$μ$ model, the $η$-$μ$ fading model as a particular case. This has important relevance in practice, as it allows for the unification of these popular fading distributions through a more general, yet equally tractable, model. The convenience of new underlying physical models is discussed. Then, we derive simple and novel closed-form expressions for the asymptotic ergodic capacity in $κ$-$μ$ shadowed fading channels, which illustrate the effects of the different fading parameters on the system performance. By exploiting the unification here unveiled, the asymptotic capacity expressions for the $κ$-$μ$ and $η$-$μ$ fading models are also obtained in closed-form as special cases.