Operating systems can run on VMs or containers. When comething runs on top of those OSs, they are not running on bare metal.
A "bare metal" display server can run in VMs, or even in containers (although the latter is a little pointless as the device access violates sandbox safety).
I'll have to say though, I find that particular definition of bare metal quite strange.
"Nested" is more or less the de-facto term in Wayland-land for the child scenario (pretty much all Wayland display servers natively support a rootful child mode either as real feature or for testing/debugging, some even under X), but there isn't really a good word for the normal case.
It's true that you don't have to, but you certainly can.
In fact, the first implementation of containers in Linux - Linux Containers, or LXC - works exactly that way.
This may or may not be as great a distinction as you might want in order to consider it bare metal vs. not. I'll leave that up to you.
├── hw
│ ├── kdrive
│ ├── vfb
│ ├── xfree86 <- this one is Xorg
│ ├── xnest
│ ├── xquartz
│ ├── xwayland
│ └── xwin
Xorg/xfree86 is the one that renders to your display hardware (metal), compared to `vfb` which renders to the Linux Virtual FrameBuffer API, `xnest` which renders to another X server, `xquartz` which renders to Apple's "quartz" display system, `xwayland` which renders to a wayland server, and xwin which renders to Win32.And, uh `kdrive`, which renders to... um... something or other.
Edit: Also, while Linux now has Atomic ModeSetting and the Direct Rendering Manager APIs, for a lot of Xorg/Xfree86's history it didn't, and the X server did a lot more direct manipulation of the display hardware than it does now. Switching virtual consoles between X consoles and Linux terminal consoles used to be a weird dance between Xorg in userspace and the kernel to make sure that the hardware handoff between them didn't mess up. This dance took about a second to complete (which is forever in CPU time) rather than being the instantaneous sort of switch it is now. But even now, with DRM and the other stuff, Xorg still does a lot of talking directly to the hardware.
What's really weird about reading this is that I can hear, in the back of my mind, the sound my CRT makes while doing this switch. But I have no memory of ever switching from X -> virtual console since LCDs were a thing. I guess... that's when X stopped crashing routinely? Weird stuff!
(Also, am I crazy or is there also a smell associated with this operation? I swear I remember something like that. When the mode switches, the monitor degausses itself, and then there is a faint smell of ozone? Computers used to be weird!)