We study the processes $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}$ and $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$ using a data sample of $519\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy ${e}^{+}{e}^{\ensuremath{-}}$ collider at center-of-mass energies at and near the $\mathrm{\ensuremath{\Upsilon}}(nS)$ ($n=2$, 3, 4) resonances. We observe ${\ensuremath{\eta}}_{c}$ decays to both final states and perform Dalitz plot analyses using a model-independent partial wave analysis technique. This allows a model-independent measurement of the mass-dependence of the $I=1/2$ $K\ensuremath{\pi}\text{ }\mathcal{S}$-wave amplitude and phase. A comparison between the present measurement and those from previous experiments indicates similar behavior for the phase up to a mass of $1.5\text{ }\text{ }\mathrm{GeV}/{c}^{2}$. In contrast, the amplitudes show very marked differences. The data require the presence of a new ${a}_{0}(1950)$ resonance with parameters $m=1931\ifmmode\pm\else\textpm\fi{}14\ifmmode\pm\else\textpm\fi{}22\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ and $\mathrm{\ensuremath{\Gamma}}=271\ifmmode\pm\else\textpm\fi{}22\ifmmode\pm\else\textpm\fi{}29\text{ }\text{ }\mathrm{MeV}$.
