In this study, we investigate the coexistence of short- and long-term memory effects owing to the programmable retention characteristics of a two-dimensional Au/MoS₂/Au atomristor device and determine the impact of these effects on synaptic properties. This device is constructed using bilayer MoS₂ in a crossbar structure. The presence of both short- and long-term memory characteristics is proposed by using a filament model within the bilayer transition-metal dichalcogenide. Short- and long-term properties are validated based on programmable multilevel retention tests. Moreover, we confirm various synaptic characteristics of the device, demonstrating its potential use as a synaptic device in a neuromorphic system. Excitatory postsynaptic current, paired-pulse facilitation, spike-rate-dependent plasticity, and spike-number-dependent plasticity synaptic applications are implemented by operating the device at a low-conductance level. Furthermore, long-term potentiation and depression exhibit symmetrical properties at high-conductance levels. Synaptic learning and forgetting characteristics are emulated using programmable retention properties and composite synaptic plasticity. The learning process of artificial neural networks is used to achieve high pattern recognition accuracy, thereby demonstrating the suitability of the use of the device in a neuromorphic system. Finally, the device is used as a physical reservoir with time-dependent inputs to realize reservoir computing by using short-term memory properties. Our study reveals that the proposed device can be applied in artificial intelligence-based computing applications by utilizing its programmable retention properties.