That's the name of the mecha that she was inside of during the first act of the second game. She didn't want anyone to know who she was and liked the name.
MULE_FEHU_EORIA_ARTONELICO is her birth name, but due to the events of the war against Eolia, she hacked her name to MULE_TEIWAZ_ARTONELICO so that her name no longer identifies Eolia as being her master copy.
"Mir" was a retarded mistranslation that completely missed the fact that humans called her "Mule" because they created her to work as one for her entire life without caring about her emotions, which they tried to erase after discarding EXEC_HARMONIUS as being useless.
EDIT:
As for Finnel, she seems interesting, but I don't know if that's enough and talking about the songs, I wanted to hurt someone after hearing EXEC_EP=NOVA.