Transparent light-converting $Y_3{\mathrm{Al}}_5{O}_{12}:\mathrm{Ce}$ (YAG:$\mathrm{Ce}$) ceramics have become promising materials for advanced light-emitting diode and laser-driven white-light-generation technology. However, their functionality is highly dependent on factors such as structural quality, uniformity of dopant distribution, the presence of defects, and the cost of fabrication. The emission of YAG:$\mathrm{Ce}$ ceramics remains thermally stable when high-power-density laser excitation is uniformly spread over the entire surface area of the ceramics due to light-scattering processes. The creation of light-scattering centers that can provide homogeneous white light in ceramics is crucial. One of the drawbacks of YAG:$\mathrm{Ce}$ as a material for converting ultraviolet emission into white light is the lack of a red component, which results in cold light and a low color rendering index. Therefore, the codoping of YAG:${\mathrm{Ce}}^{3+}$ with red-emitting ions (${\mathrm{Er}}^{3+}$, ${\mathrm{Ho}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Pr}}^{3+}$, and ${\mathrm{Cr}}^{3+}$) was carried out and slight nonstoichiometry was induced in the present study to improve the photoconversion parameters. The phase purity of the studied ceramics was analyzed using powder x-ray diffraction and SEM combined with energy-dispersive spectroscopy (EDS). EDS studies revealed that nonstoichiometric ceramics contained ${\mathrm{Al}}_2{O}_3$ phase inclusions that acted as light-scattering centers. Optical transmittance and absorption spectra measurements revealed the presence of light-scattering centers imposed by ${\mathrm{Al}}_2{O}_3$ phase inclusions and nanodefects related to perturbations of the local lattice structure by dopants. EDS mapping and photoluminescence analysis showed the formation of $\mathrm{Ce}$-$\mathrm{Ln}$,$\mathrm{Cr}$ ($\mathrm{Cr}$ is a transition metal) pairs and the availability of energy transfer between the pair-forming ions. Intrinsic thermal quenching of ${\mathrm{Ce}}^{3+}$ luminescence started above 400 K. The nonstoichiometric YAG:$\mathrm{Ce},\mathrm{Nd}$ sample showed luminescence stability up to 650 K with a luminous efficacy of 198 lm/W. The photoconversion parameters of the transparent ceramic packaged with a blue light-emitting diode or a laser diode were compared. It was shown that the color temperature of ceramic emissions could be controlled by the spectral width of the excitation beam. Thermally stimulated luminescence was applied to investigate the deep trapping centers imposed by the codopants. A detailed EPR study revealed the presence of $F^{+}$, $O^{-}$, and ${\mathrm{Ce}}^{4+}$ centers.