The translation says that the reincarnation is a one-time-use ability, and it was spent on MC 200 years ago, and Alice was registered as a wielder of the sword one year later. That's it.
Ever since the princess was introduced in chapter 7, I'd always thought that Alice had suffered a cruel fate at the hands of the kingdom. She'd be captured as a witch, and for gaining control of her powers, she'd be forcibly married into royalty. That, OR, she was kept as a slave in some basement, and similar to Ymir in Shingeki no Kyojin, she'd be raped by the king over and over until a heir was produced, after which she was killed. Thus, cutting any connection to the Feera village..
BUT.. you can re-read chapter 13 for the princess' explanation about her family circumstances, as she knows it from hearsay. The unnamed predecessor of her grandmother, who lived for who knows how long, was most definitely hinted as being Alice.