The transport and synaptic characteristics of the two-terminal Au/Ti/ amorphous Indium-Gallium-Zinc-Oxide (a-IGZO)/thin SiO2/p+-Si memristors based on the modulation of the Schottky barrier (SB) between the resistive switching (RS) oxide layer and the metal electrodes are investigated by modulating the oxygen content in the a-IGZO film with the emphasis on the mechanism that determines the boundary of the abrupt/gradual RS. It is found that a bimodal distribution of the effective SB height (ΦB) results from further reducing the top electrode voltage (VTE)-dependent Fermi-level (EF) followed by the generation of ionized oxygen vacancies (VO2+s). Based on the proposed model, the influences of the readout voltage, the oxygen content, the number of consecutive VTE sweeps on ΦB, and the memristor current are explained. In particular, the process of VO2+ generation followed by the ΦB lowering is gradual because increasing the VTE-dependent EF lowering followed by the VO2+ generation is self-limited by increasing the electron concentration-dependent EF heightening. Furthermore, we propose three operation regimes: the readout, the potentiation in gradual RS, and the abrupt RS. Our results prove that the Au/Ti/a-IGZO/SiO2/p+-Si memristors are promising for the monolithic integration of neuromorphic computing systems because the boundary between the gradual and abrupt RS can be controlled by modulating the SiO2 thickness and IGZO work function.