Low-spin states in ${}^{124,126,128,130}\mathrm{Ba}$ fed by the $\mathrm{EC}/{\ensuremath{\beta}}^{+}$ decay of their La parents have been investigated by means of the \ensuremath{\gamma}-\ensuremath{\gamma} angular correlation measurement coupled with an isotope separator on-line. The spin of the first excited ${0}^{+}$ states ${(0}_{2}^{+})$ were unambiguously established, and higher-excited ${0}^{+}$ states were newly identified. The energy of the ${0}_{2}^{+}$ state in ${}^{124}\mathrm{Ba}$ previously assigned was revised upward. Resultingly, the level energy of the ${0}_{2}^{+}$ state in neutron-deficient Ba nuclei takes a minimum at $N=72$ and then gradually increases toward neutron midshell $(N=66),$ while the level energy of the ${0}_{3}^{+}$ state rapidly decreases with decreasing neutron number. From an extrapolation of these ${0}^{+}$ energies, it is highly expected that the energy relation between the ${0}_{2}^{+}$ and ${0}_{3}^{+}$ state would invert at ${}^{122}\mathrm{Ba}$ or at more deformed Ba nuclei. This energy inversion is interpreted as the evolution of the ${0}_{2}^{+}$ state in \ensuremath{\gamma}-soft nuclei toward the two-phonon \ensuremath{\gamma}-vibrational ${0}^{+}$ state in axially-symmetric deformed nuclei, while the ${0}_{3}^{+}$ state toward the \ensuremath{\beta}-vibrational ${0}^{+}$ state.