I think part of the reason Haley never bonded with Undone (other than what she said in interviews, that it's
not something she personally had experienced at the time it was written-that level of heartbreak)- is that right when it could have taken off due to movie exposure and fan approval, she was going through her own heartbreak. Breaking up with her label and the dream they would support her creativity, and a possible boyfriend breakup (Stefano). I think she didn't like singing that song because she's a very strong woman who identifies more with "I threw your ring into the ocean" (see Tusk, I know where you're coming from too), rather than heartbreak and "nothing but emptiness inside".
I just think there's a chance for her to convey that tenderness and pulling of heartstrings of love and love lost in both songs, which she does beautifully with that amazing ability to go between regular singing and head voice / falsetto, it's just perfect for a ballad to convey crying a tear and a love laid bare whether it's through heartbreak or falling in love. Her warm honey whiskey voice is so so good for ballads.