What if I run linux + gnu + gnome over rdp on VM a server in a rack somewhere that has no screen keyboard and mouse on it? Am i using desktop linux or not?

What if that same VM also is running nginx and serving up web content?

What if I have a pc with a keyboard and monitor sitting literally on my desktop, and it's running linux + gnu but no graphical environment, and I use it for coding (it has music playing when I do this, and i sometime check email or github issues, etc via cli) - yes I've done this, even recently to reduce distractions... some days GUIs are bad for my adhd. Is that a desktop linux? If not, why? What's different about this than doing basically the same thing, but also having a browser open when it's surrounded by a GUI?

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.