I’m clearly weak on terminology. When you say Synapse instance, Sypnase is the server software that would then allow me to host an instance of Element/Matrix? And an instance would be running that software somewhere — namely, a device I own or a VPS?
Yes. Synapse is the server side part of Matrix. You install it at a Matrix host, or on a VPS, or even a Raspberry PI device. See https://github.com/matrix-org/synapse#installing-and-configuration
Once up and running you connect a client like Element to it.