Sometime between D1 and D2 Billie goes into Stiltons home, gets caught by guards and loses her eye and arm, which we see in Dishonored 2. If during the 'crack in the slab' you knock Stilton out, thus preventing him from going to the ritual and losing his mind, when you return to the present the entire manor is different because he is still sane and has protected the people of the Dust District. Because of this, for the rest of the game Billie has her eye and arm, which is the canon ending. This creates a rift in time that the Outsider corrects, by taking her arm and eye and replacing them with void relics, making it so she simultaneously does and does not have an eye and arm, which "fixes" the time problems. Her powers are not tied to the Outsider at all, hence the new mana meter and lack of runes. She can't use runes to improve her control over the void because she doesn't draw her power from the Outsider. This also means that when the Outsider is either killed or spared, she would in theory keep her powers.
"Destroy the rats! They carry secrets as well as diseases." - Witch
"People usually want something from me - but you don't want my Mark, do you? So I will return something you lost." -Outsider
"You're not even some horror I need to face down and fight. Your at my mercy. But you've always been at the mercy of bad people, haven't you? You don't deserve this. No one deserves this."' -Billie (to Outsider)