I expect web apps to auto-save because writing text in browsers is historically brittle, because browsers don't naturally persist the things you type into them anywhere. So if you don't create some layer of persistence, there is no natural point at which the user expectably saves things. Auto-save is just part of creating reliable, persisted data in web apps (regardless of whether that is local or remote) without expecting the user to do anything. There is no standard for manual saves to rely on, like there is for native applications that save to files.

So yes, you're old fashioned.

I still hit "Save" in every single offline or online app every time I've made a minimal change, because of old-timer PTSD reflexes: I used to edit files in vi over modem and telnet without screen (tmux), and if the connection had a hiccup, the connection died, my editor died, and the changes died with it. So saving (extremely) frequently was just a way to not lose your changes.