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Nonlinear light-matter interaction at low energy, particularly in the terahertz (THz) frequency range, hosts unique phenomena distinct from the optical excitation with photon energy of a few eV. In cuprate superconductors Bi2Sr2CaCu2O8+x, the THz nonlinear response is identified via the optical reflectivity change and interpreted as the amplitude mode of the superconducting condensate, namely the Higgs mode [K. Katsumi et al., Phys. Rev. Lett. 120, 117001 (2018)]. However, the origin of the THz nonlinearity has been questioned because the pair-breaking process, identified in Raman spectroscopy, can also contribute to it. Here, we reexamined the THz-driven nonequilibrium dynamics in cuprates Bi2Sr2CaCu2O8+x by comparing it with the Raman susceptibility. In the optical reflectivity change, we found an oscillatory behavior following the squared THz waveform (THz Kerr signal), as well as the relaxation of the quasiparticle excitation. Careful insight into the data revealed that the oscillatory and decaying contributions exhibit different doping dependence. Remarkably, the doping and temperature evolutions of the THz Kerr signal are distinct from those of the Raman susceptibility, which is described by the pair-breaking due to diamagnetic light–matter interaction. These results indicate the importance of the paramagnetic light–matter coupling in the THz Kerr signal in the cuprate superconductors, likely arising from the Higgs mode.