For clarification~

Lucia is her childhood name, a name her mom called her by

Vivian is her "proper princess name" and hue knew her as Princess Vivian so he calls her Vivian

Lucia is her actual name and Vivian is the name the King gave her when she first began living in the palace. Those who she holds really dear to her calls her Lucia.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
This causes a huge problem later

Case of double identity? She is called Vivian cause that is her princess name. She doesn't like it because it's associated with all the things wrong that happened in her past life. Lucia is the name she chose (i think?) that is associated with her not being a princess and more free (like after getting away from that horrible marriage in her past life she was able to live her own life as 'Lucia'). This is totally just my theories but there probably other reasons why too. :P

And I don't think Hue knows that she is also called Lucia, i think he only knows her as Princess Vivian.

Spoiler.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
Her mother gave her that name. When she escaped she just went back to using that name. Hue gets really jealous of people calling her Lucia. Especially when Damien calls her Lucia instead of Vivien. Hue begins to think she does not like him all that much because she never allowed him to call her Lucia. It later gets resolved tho

Hehe I know... （TДT) i read the novel but it got confusing after rereading it ╮(￣▽￣)╭