I feel like you're overthinking it. It's not that one can get a badge saying "powered by Desktop Linux". It's a rough categorisation based on the use case:

* Embedded Linux is what you expect to see on a "small" device that usually doesn't have a graphical environment (it may have a small screen showing a temperature).

* A Linux server is what you expect to see in racks, serving stuff over the Internet. A homeserver could be that, too.

* Linux on mobile is what you would put on your phone.

* Desktop Linux is what you would put on your working computer, the one you interact with "physically".

Of course, you can run a server on your personal laptop, and you could run a "Desktop" graphical environment on a mobile phone. But that's beside the point. And of course, you can work on a Linux without a graphical environment.