Just gone back to look for those possible census entries I found for Peter.
There is this familly in 1861:
Peter Waters 27, agr. lab. b.Ireland
Ann Waters 25, b. Ireland
Mary Waters 6 b. Ireland
James Waters 4 b. Ireland
Thomas Waters 2 b. Ayton, Berwickshire
Hannah Waters 1 b. Mordington, Berwickshire
Helen Boice 15, niece b. Ireland
Address: Bastling Farm Cottage "rd", Ayton, Berwickshire
From IGI births, I think mother Mary's maiden name looks to be Boyce which fits with the surname for the niece.
1871:
Peter Waters 46, labourer b. Ireland
Martha G Waters 40, b. Livingston, Linlithgow
Hannah Waters 12 b. ??, MacCington (as transcribed)
Peter Waters 10 b. ??, Ayrton
William Waters 1 b. Livingstone
Address: No 24 West End, Livingstone, West Lothian
Mother for young William above shows as Martha Gallagher. Likely second marriage for Peter Snr.?