‘Lucia’ was the name given by her deceased mother, while ‘Vivian’ was the name given by her father (the current king/emperor, who is also Kwiz’s (the Crown Prince’s) father).
After Lucia ran away from Matin’s mansion, she met an old woman (who was mentally disturbed) that lost a daughter (whose name was ‘Lucy’) & lived with that woman as her daughter. Then after the woman died, Lucia went to the local government & said that there’s a mistake in writing her name (as ‘Lucy’) & she asked to change that name into ‘Lucia’, thus she gained her original name back