The classification of dark matter halos as isolated hosts or subhalos is critical for our understanding of structure formation and the galaxy-halo connection. Most commonly, subhalos are defined to reside inside a spherical overdensity boundary such as the virial radius. The resulting host-subhalo relations depend sensitively on the somewhat arbitrary overdensity threshold, but the impact of this dependence is rarely quantified. The recently proposed splashback radius tends to be larger and to include more subhalos than even the largest spherical overdensity boundaries. We systematically investigate the dependence of the subhalo fraction on the radius definition and show that it can vary by factors of unity between different spherical overdensity definitions. Using splashback radii can yet double the abundance of subhalos compared to the virial definition. We also quantify the abundance of flyby (or backsplash) halos, hosts that used to be subhalos in the past. We show that the majority of these objects are mislabeled satellites that are naturally classified as subhalos when we use the splashback radius. We compare our results to self-similar universes to show that the subhalo and flyby fractions are not universal with redshift and cosmology.