While I hesitate to agree with that definition of local-first, I agree with your notion thats what the author by large is talking about and unfortunately many such an web-app have the worst of both worlds.

Most uses of the term local-first I see regularly mean to say, "an app that doesnt require 3rd party servers outside of your control to function", within some level of reason. Sometimes agnostic to how it's data is synced, sometimes meaning its an exercise left to the user entirely, sometimes meaning fully self-hosted and hardly local to the device being used to access the app.