In this context, however, there's an additional expectation that these devices make this happen over a network, so that you can handle things like BIOS configuration fully remotely.
Yeah, the important distinction here is that these devices somewhat fulfill the role of a standalone IPMI[1] device. IPMI is typically built into server motherboard (SuperMicro) which allows an administrator to reboot a machine or configure BIOS over a network. Your typical KVM does not feature this.
However, where these lack is that they still require a video card. IPMI built into a motherboard can run on an entirely headless system.
There are also commercial versions of this, such as the Lantronix Spider KVM over IP[2]
[1] https://en.wikipedia.org/wiki/Intelligent_Platform_Managemen...
Edit: I say these here mainly because expanded out, the title says "Raspberry Pi 'Keyboard Video Mouse's Compared". That still means nothing to someone who doesn't know what a KVM is (actually makes it sound like HID devices specifically made for the Pi).