A single "server" (one physical computing machine) could have hundreds of "servers" (instances of the server-side of the game software) running on it. One physical tower PC could have "10 Quake Servers", "2 HL2-DM servers", "a Team Fortress 2 server", "a Unreal Tournament server" and such on it.
Discord originally seemed to be using that era of lan-party style language for it's naming conventions. A Discord "server" is just an instanced section of the server-side of the Discord software, and not specifically related to any real-world physical machine.
By contrast, only Discord-the-company can run Discord servers.
The exception, of course, being games for which it is actually still possible to run your own servers. Minecraft comes to mind.
> recent games
You absolutely would. Just because maybe you're playing overwatch or league or something, doesn't mean this isn't still a hugely popular paradigm.
Contrast with Discord where there's something of an illusion of ownership; people will refer to them as "my Discord server."
EDIT: Incidentally, if you're actually running a private server for an MMO, I would indeed call that "your server."
It doesn't matter how temporally limited it is, it doesn't matter if it's on dedicated hardware or not, or how many "servers" are sharing that hardware. It doesn't even have to be on a different machine.
> Typically that list of servers in online games was actually a list of different physical locations you could connect to.
And? If you saw me hosting two cs lobbies from the same box, would you stop calling them servers?