I've just done a bit of scavenging myself on Scotlandspeople, and Jane Renton turns out to be full of surprises.
Monica gave you the line-up for Jane and her family in Oldhamstocks parish in 1881. Jane was then 51. Believe it or not, she married a Thomas Robertson two years later, in 1883. Thomas was 59 and a widower, Jane said she was 52 and a widow. She gave her parents (deceased) as John Renton, shoemaker, and Elizabeth Anderson.
I wondered if she really was a widow, and if so, why was she using her maiden name of Renton? So I did a check on the IGI, and in fact Jane Renton married a George Renton(!) in 1865 at Oldhamstocks. She was then 34 and he was 60. So she clearly preferred older men.
This marriage-certificate gives George's parents as Thomas Renton, carrier, and Janet Sheriff, and Jane's parents as George[sic] Renton, shoemaker, and Elizabeth Anderson. I think this is just a clerical error for John.
In 1871 George and Jane were living at Fernylea cot house, parish of Oldhamstocks, and also living with them was Agnes Peebles, "step-daughter".
I'm still puzzled as to why the fishing family of Watson/Cowe in Cellardyke, Fife, but hailing from Northumberland and/or Durham, had a daughter called Jane Renton Watson.
Harry