If you said "I'm carless" and I disagreed because you'd moved house or used a taxi, you'd think I was being a little odd.
These terms make much more sense to me because they describe what I care about. You pay someone to get you from A to B. They do so in a car, but it wouldn't matter if you sat in the same seat with the same experience but was on a bus, or the car-shaped box you were in was actually sitting on a train moving you along. That's a detail you don't care about.
I want to run code. I don't care if it's my server, a shared one, it moves from server to server or starts and stops on demand. I don't care about the server. I don't want to care about the server part.
I get the annoyance, but I think they're useful.