Does an OS necessarily need to function without being connected to the Internet? Is there anything wrong with a definition of OS that includes "Internet connectivity"? How does Internet connectivity differ from hard disk connectivity, RAM connectivity, floppy disk connectivity, etc? That brings up questions of what the word "function" means. My phone will technically function without any wireless connection, but then it's not really a phone, is it? The Echo will turn on without an Internet connection, at least far enough to be able to say "I'm sorry, I don't have an Internet connection" most likely.

It's the ages old question: is Linux an OS? No, it's a kernel. Is Ubuntu an OS? No, it's a distribution. Is Gnome an OS? No, it's a desktop environment.

You could say with a straight face that Alexa is an OS that's designed to run cloud software. The OS works without a connection, but the user land might not.