The family looks to have been living in Co. Cavan. Here’s Peter’s death in Tullyroane, Cavan in 1898. The informant was son in law James MacAulay of Killoughter.
I think that after Peter’s death Rebecca moved to Scotland to be with her daughter and son in law. She died in Law on 11.11.1901 aged 66. Lizzie was the informant. Rebecca’s parents names are not on her death certificate (the section has a blank line drawn through it), indicating Lizzie didn’t know who they were. So possibly Rebecca was an orphan or adopted. Her origins may be hard to trace.
Macaulay family are in the 1901 Scottish census Carluke district, 629/4/43. Parents plus 3 children (William, Richard & James). Rebecca doesn’t appear to have been there though.