Revert "Remove the trigger for the first-login mode"

This reverts commit f2bc7d626e14f0c75b554a1269fca3b61a799073.

Fedora is one of the main users of gnome-initial-setup, and its
installer still supports creating the user ahead of time. As long
as thats the case, removing this functionality causes a regression.

This went unnoticed in 3.12 since there was no Fedora release with
it.
Explicitly use existing-user mode in the session

Since we default to new-user mode now, use the --existing-user
option when running inside the session for a preexisting user.