It's "serverless" in the sense of the server being commoditized and irrelevant to the design of the app. It might be better to say that the server is rendered ubiquitous or trivial than that it's removed. Lambda gives you a programming model where the server has no configuration or state, because all of what would normally be in the server is now in your handler specifications instead. So you don't need to think about the server, and any server running anywhere that can support loading Lambda handlers would be equivalent for your purposes.

Imagine you're running your own Lambda-compatible server, and you have a scaling problem. The programming model means that the solution-space to that problem is restricted to adding more servers in a shared-nothing cluster configuration, not to reconfiguring the ones you have. Your Lambda server is thus a piece of black-box infrastructure, like a load-balancer.

I think that understates it a bit. AWS Lambda is backed by a fleet of servers managed around the clock by one of the largest and most experienced IT organizations in the world, and API Gateway is fronted and cached by Amazon CloudFront's global network of servers.

It's shared hosting all over again!

We need a new term, because "someone else's server" is a superset of what's happening here. How about "transient server" or "one-shot server"?