It's on the server side, but you don't manage any servers.

Serverless means you don't manage any servers, directly or indirectly. It's a step further towards where you don't own any servers.

It's part of an evolution that has gone from being more aware of servers to being less aware: on-site servers -> colocated physical servers -> dedicated servers -> virtual private servers -> cloud virtual private servers -> autoscaled cloud VPS -> cloud functions where you you aren't billed by, or even aware, of the number of servers running

I don't think "serverless" is a terrible term.