Multiple-access edge computing (MEC) is an essential means of improving network infrastructure performance and an important development direction for future satellite-terrestrial integrated networks (STIN). However, migrating existing MEC technology on the ground directly to satellites is challenging. On the one hand, the performance of terrestrial network elements is much larger than that of satellite nodes, and satellite functions need to be customized. On the other hand, owing to the mobility of low Earth orbit/medium Earth orbit satellites, the provision of reliable services still needs to be studied. This paper proposes MEC-based STIN architecture to address these challenges in different ways. Considering the deployment of UPF and other network elements, STIN architecture significantly impacts resource utilization. These architectures consider the technical characteristics of different MEC deployments, and a simulation system is constructed to test the architectural performance.