i think it still may be possible that we're related, maybe there is a cousin connection.
this is the info i have on my rainsford (1911 census)
RAINSFORD, John Henry Husband Married M 42 1869 Dock Labourer Kilcavan Co Wicklow Resident
RAINSFORD, Ann Wife Married 15 years F 36 1875 Oddington Gloucestershire
RAINSFORD, John Henry Son M 14 1897 Errand Boy Liverpool Toxteth Park Lancs
RAINSFORD, James Patrick Son M 12 1899 School Liverpool Toxteth Park Lancs
RAINSFORD, Edward Joseph Son M 10 1901 Liverpool Toxteth Park Lancs
RAINSFORD, Mary Daughter F 9 1902 Liverpool Toxteth Park Lancs
RAINSFORD, Lawrence Son M 7 1904 Liverpool Toxteth Park Lancs
RAINSFORD, Francis Son M 5 1906 Liverpool Toxteth Park Lancs
RAINSFORD, Ann Daughter F 3 1908 Liverpool Toxteth Park Lancs
RAINSFORD, Thomas Son M 1 1910 Liverpool Toxteth Park Lancs