Regarding the Louisa/Eliza on the 1881 census- I wonder if there has just been a mishearing of the names? Children's names more or less match as you say - except that James should be Joseph my Great Grandfather!
I'm looking back at all the various records I have.
So in 1891 census I have Joseph (born Liverpool) aged 15 (who I believe to be my great grandfather) living with his father Patrick, a cooper, (widower), age hard to make out but could be 47 or 52.
When Joseph marries in 1899 aged 23 his father is listed as Peter Cain - not marked as deceased - and mother Eliza Cain ms Cullen deceased.
1901 census has Joseph Cain - definitely the right one! - aged 25 born England, a boilermaker, married by this time to Isabella Duncan. Has various children including my grandmother so can say with confidence this is him!
Patrick Cain (or Kane)'s death in 1902 has him aged 57, a cooper, widower of Elizabeth Cullen.
I think I need to find Joseph's brothers and sisters records and see what I can get from them. thanks everyone for your help so far. Another trip to the records office required!