Yes: they're talking about a computer running a desktop environment on Linux on a desktop or laptop computer. ChromeOS meets all those criteria. Android would meet all those criteria should it ever be widely deployed on laptops or desktops.
So:
"It's a simple question: at what point do you differentiate 'it runs on Linux' from 'it's Linux on the desktop'?"
And it's a simple answer:
1: It's a desktop or laptop computer.
2: It's running Linux.
3: It's running a graphical desktop environment.
If all three of those things are true, then it's Linux on the desktop. All three of those things are true for Ubuntu, so it's Linux on the desktop. All three of those things are true for ChromeOS, so it's Linux on the desktop.
Whether the operating system allows lower-level access (like a command-line interface) is irrelevant.
"Because if my router counts, Linux won a long time ago."
Your router probably doesn't meet those criteria, since it's probably not a desktop computer and probably not running a desktop environment.
"Android smart TVs?"
Fails criterion 1.
"Routers?"
Fails criteria 1 and 3.
"Mall kiosks?"
Probably fails criterion 3, and almost certainly fails criterion 1.
"Car infotainment systems?"
Fails criterion 1.
"Bluray players where the only interface is a Java program?"
Fails criterion 1.
