Did a bit of digging; the first client gets forked to create the "server". The forked server then detaches and runs in the background. You're right that -x creates an entirely new, separate client process, unrelated to the OG client or the forked server.

Without -x though it works as originally described.

Edit: gnu screen 1.0 was originally released in 1987. The -x flag was released in screen 3.0 in the 90s. TIL