We report on the first proton-induced single proton- and neutron-removal reactions from the neutron-deficient $^{14}$O nucleus with large Fermi-surface asymmetry $S_n-S_p$ = 18.6 MeV at $∼$100 MeV/nucleon, a widely used energy regime for rare-isotope studies. The measured inclusive cross sections and parallel momentum distributions of the $^{13}$N and $^{13}$O residues are compared to the state-of-the-art reaction models, with nuclear structure inputs from many-body shell-model calculations. Our results provide the first quantitative contributions of multiple reaction mechanisms including the quasifree knockout, inelastic scattering and nucleon transfer processes. It is shown that the inelastic scattering and nucleon transfer, usually neglected at such energy regime, contribute about 50% and 30% to the loosely bound proton and deeply bound neutron removal, respectively. These multiple reaction mechanisms should be considered in analyses of inclusive one-nucleon removal cross sections measured at intermediate energies for quantitative investigation of single-particle strengths and correlations in atomic nuclei.