Not necessarily. Back in the serverless CGI days you would still typically manage Apache. What you say is probably why would you choose serverless in 2025, but not always. This is not the technical differentiation.

> There is still a server running my app

There may be a separate server (e.g. Apache) that runs your app, but that is someone else's app, not your app. Calling someone else's app your own would be rather silly. Your application is less a server, which is a change from how these applications had normally been written over the past couple of decades. At least since the death of CGI, you would traditionally see your application become the server (possibly proxied through another server, but that is ultimately immaterial to the discussion). That is what goes away with serverless.