Bismuth-activated light emissions have been much reported in ternary alkaline earth ${d}^{10}$ metal oxides, and the self-activation nature of these hosts makes it possible to achieve efficient luminescence via energy transfer between the host and the activator ions. A detailed analysis of the potential intrinsic defects accounting for the intrinsic emissions is still absent, and the identifications of bismuth-related transitions remain unclear or controversial. Given these facts, first-principles calculations using hybrid functionals are employed to systematically study the bismuth-doped $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Sb}}_{2}{\mathrm{O}}_{6}$, ${(\mathrm{Sr}/\mathrm{Ca})}_{2}{\mathrm{Sb}}_{2}{\mathrm{O}}_{7}$, $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{SnO}}_{3}$, and $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Ga}}_{2}{\mathrm{O}}_{4}$, which contain ${d}^{10}$ metals belonging to VA, IVA, and IIIA groups, respectively. Primary intrinsic defects and self-trapped holes in these hosts are revealed and the roles played by them on intrinsic emissions are evaluated. Bismuth ions are identified as trivalent dopants and their $^{3}P_{0,1}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}^{1}S_{0}$ (referred to as $A$-band) transition is assigned to account for the ultraviolet emission observed in bismuth-doped $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Sb}}_{2}{\mathrm{O}}_{6}$, while the visible emission observed in $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Sb}}_{2}{\mathrm{O}}_{6}$:Bi is attributed to ${\mathrm{Bi}}^{3+}$ dopants in the minute amount of ${(\mathrm{Sr}/\mathrm{Ca})}_{2}{\mathrm{Sb}}_{2}{\mathrm{O}}_{7}$ impurities. In ${\mathrm{Sr}}_{2}{\mathrm{Sb}}_{2}{\mathrm{O}}_{7}$ and ${\mathrm{Ca}}_{2}{\mathrm{Sb}}_{2}{\mathrm{O}}_{7}$ with bismuth, the excitation bands are identified as $^{1}S_{0}\ensuremath{\rightarrow}^{3}P_{0,1}$ transition of ${\mathrm{Bi}}^{3+}$, and the observed emission bands are attributed to charge transfer transition (${\mathrm{Bi}}^{2+}+{h}_{\mathrm{VBM}}^{+}\ensuremath{\rightarrow}{\mathrm{Bi}}^{3+}$) and $A$-band transition ($^{3}P_{0,1}\ensuremath{\rightarrow}\phantom{\rule{4pt}{0ex}}^{1}S_{0}$), respectively. Furthermore, the bismuth-related transitions are also well identified in $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{SnO}}_{3}$ and $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Ga}}_{2}{\mathrm{O}}_{4}$ and the large Stokes shifts of the $A$-band transition in $(\mathrm{Sr}/\mathrm{Ca}){\mathrm{Ga}}_{2}{\mathrm{O}}_{4}$ are revealed and analyzed. The insights gained in this work extend our understanding of the properties of intrinsic defects and the luminescence mechanisms of insulating bismuth-doped ternary oxides.