In this work, we carry out a first-principles investigation of intrinsic paramagnetic point defects in P$_2$O$_5$ and in Na$_2$O−P$_2$O$_5$ glasses as a representative of alkali phosphate glasses. Glass models are generated by combining classical molecular dynamics and Monte Carlo simulations and validated by comparing their corresponding structure factors with the available X-ray and neutron scattering experiments. We use density functional theory to calculate the electron paramagnetic resonance parameters for a large set of paramagnetic oxygen-vacancy configurations. Our investigation, also by unveiling the effect of the local environment and disorder on the hyperfine tensor, enables us to propose a new model for the much debated P$_3$ center. In particular, we establish the occurrence of two variants, which we name P$_3^a$ and P$_3^b$ centers, that are instrumental to explaining the experimental shifts of the hyperfine splittings observed in alkali phosphate glasses as a function of the alkali content x in the phosphate glass. Our scenario predicts that for low to intermediate alkali contents (0 < x < 50%), a mixture of P$_1$ and P$_3^a$ centers should be generated under irradiation. For x > 50%, essentially only P$_3^a$ and P$_3^b$ centers would be generated, while P$_1$ will be absent. Therefore, our findings, by providing an improved mapping of P centers in phosphate glasses, pave the way for fine-controlling/tuning the optical absorption in a wide range of technological applications.