I'm having trouble putting a terminal from Halo 3, Halo Anniversary, and Halo 4 together. In one of the Halo 3 terminals, it's implied that the Librarian is on Earth in the shadow of Mount Kilimanjaro when the rings activate. It is also stated that she destroyed or deactivated any method she had of reaching the Ark. She is talking with the Didact up until the ring fires. In anniversary, we see the silhouette of someone sitting by Mount Kilimanjaro when the rings are activated, who promptly disappears when the ring fires. In Halo 4, we see a terminal in which the Librarian shoots and imprisons the Didact, presumably before the rings fire. So did the Librarian survive, or is this an inconsitancy.
There are two Didacts: the one in Halo 4 is the Ur-Didact, the original, the one the Librarian talks to is the Bornstellar-Didact, another forerunner who was imprinted with Didact's memories and personality.
The Librarian strands herself on Earth some time after imprisoning the Ur-Didact - she says in the Terminal that "in the days to come, the Halo rings will fire," so the events of Halo 3's Terminals where she converses with the Bornstellar-Didact play out normally.