"Stateless" is probably clearer. "Serverless" usually refers to services from cloud providers where your code is run on demand rather than having a server that is idling around (to use a sibling comment's phrase). I don't really care for "serverless" as a term because there are still servers (as you point out) and there are other architectures and cloud services that seem equally serverless that aren't usually called that.

serverless is when you save the state but don't keep a machine sitting around when there are no requests, so you'd keep your calendar state in a hosted db and when someone asks for the calendar, a thread spins up and creates the calendar from the state and then disappears until there is another request. This is basically the exact opposite of serverless

Isn't it serverless when it's local-only? Or, in an extreme case, peer-to-peer?

Your app is always running to answer those GET so it's not serverless. Serverless means that your rust code would be invoked upon GET.