Why is Riven called the Exile when she is not actually an exile?

Noxus never exiled her, because they think she is dead. Ionia didn't exile her either. They accepted her as a part of the community. According to "Confession of a Broken Blade", even Riven does not consider herself as an exile. Her biography says she exiled herself, and I understood that as she punished herself and detached her from the world. But that is no longer the case after she's forgiven by Yasuo and accepted by other Ionians (including her step-parents). Basically, her "exile" is over. Then, why is her title still "the Exile"?

Couldn't it be something else like "the Broken Blade" or "the Atoner"? It would be more accurate and fit her story.

edit. for the reference here is a part of the text from the story

“A farm in Trevale.” Riven looked at the old man. “Noxus,” she admitted. The council hall, which had dropped back to silence in order to hear the prisoner, took in a collective breath. “I see,” continued the judge. “And you no longer call that place home.” “When your home tries to kill you, is it still home?” “You are an exile then?” “That would imply I wish to return,” Riven said. “You do not?” “Noxus is no longer what it once was.” Impatience edged into Riven’s voice. “Can we get on with this?”