People still use our old-fashioned Unix login servers

So it seems like a "login server" is simply any server you can log in to?

In this context, it is a Unix machine configured for general purpose use (so no minimal web-server-centric OS, for example) where everybody in the organisation gets a login (probably automatically and tied to whatever SSO they use). There may also be some niceties installed and/or preconfigured - like software used frequently by this and that research group or auto-mounting the user's file share from the central storage cluster.

I started with UNIX in 1993 with Xenix, have used most well known commercial flavours since then.

When I started working, our PC were thin clients for all practical purposes, we had development servers where everyone logged in to work, via telnet or remote X session.

Never ever did I heard about login server term as it is being described here.

I’ve seen this setup in many academic institutions - sometimes also called a jump server too. It handles auth to the rest of the intranet and HPCs, as well as storing the user’s common data like docs, code and dotfiles - shared across all subsequent ssh hosts.