Can someone explain me how this is athy's third reincarnation ?

It's here third life, but the second reincarnation.
Her first life was as the original Princess Athanasia who gets killed by Claude. I
Her second life was as an orphan in Korea, and she remembered her first life as a story in a book (we don't know whether it was an actual book, or if she only remembered it as a book)
Her third life is now, where she remembers her life in Korea and her "book" life as the previous Athy.

Ohk thanks for clearing that up.. I was really confused