Reliably interpreting spectra from electron-ionized cosmic plasmas requires accurate ionization balance calculations for the plasma in question. However, much of the atomic data needed for these calculations have not been generated using modern theoretical methods and are often highly suspect. This translates directly into the reliability of the collisional ionization equilibrium (CIE) calculations. We make use of state-of-the-art calculations of dielectronic recombination (DR) rate coefficients for the hydrogenic through Na-like ions of all elements from He up to and including Zn. Where measurements exist, these published theoretical DR data agree with recent laboratory work to within typically 35% or better at the temperatures relevant for CIE. We also make use of state-of-the-art radiative recombination (RR) rate coefficient calculations for the bare through Na-like ions of all elements from H through to Zn. Here we present improved CIE calculations for temperatures from 10⁴ to 10⁹ K using our data and the recommended electron impact ionization data of Mazzotta et al. for elements up to and including Ni and Mazzotta for Cu and Zn. DR and RR data for ionization stages that have not been updated are also taken from these two additional sources. We compare our calculated fractional ionic abundances using these data with those presented by Mazzotta et al. for all elements from H to Ni. The differences in peak fractional abundance are up to 60%. We also compare with the fractional ionic abundances for Mg, Si, S, Ar, Ca, Fe, and Ni derived from the modern DR calculations of Gu for the H-like through Na-like ions, and the RR calculations of Gu for the bare through F-like ions. These results are in better agreement with our work, with differences in peak fractional abundance of less than 10%.