State-of-the-art memristors based on 2D transition metal dichalcogenide material as an active area formed by vertical or lateral structures are promising devices for emulating artificial synaptic behavior. They owe their switching ability to the defects of said active area. Such layers prepared by chemical vapor deposition are the most employed since they contain vacancies and grain boundaries. However, not much is said about the exfoliated active areas. In this work, we demonstrate vertical memristors based on exfoliated molybdenum disulfide, which reveals a gradual resistive switching mechanism based on Schottky barrier modulation. The devices operate without a forming step and show a gradual resistive switching behavior. The mechanism is attributed to charge trapping/detrapping, which modulates the Schottky barrier at the MoS₂/metal interface. Furthermore, the device demonstrates key synaptic functions including potentiation, depression, and spike-amplitude-dependent plasticity, highlighting its potential as a synaptic building block for analog neuromorphic computing systems.