1881 census has a Jane Redfern born Little Hay Staffordshire not Derbyshire.
Flying due south from Lichfield (Staffs) towards Birmingham, Little Hay is 4 miles, Sutton Coldfield (Warks) is 8 miles, Erdington is 10 miles.
Little Hay was a hamlet in Shenstone parish, which was in Lichfield registration district.
There's a birth in that district in q4 1874. Nothing else at all close. She's probably the Jane in the 1881 census.
Whether she's Josie from Erdington, well that still takes a bit of explaining. Maybe Josie wasn't really a proper legal wife. Doesn't matter much as the mother's maiden name is Redfern on all 4 kids.
Haven't checked 1891. Really you need the father's name on the marriage cert to pick the birth.