I don't know if you know, but in the time period of John Baines snr's marriage, Jenny was often a pet name for Jane - I've had this happen twice in my tree, once where the baptism name was jenny and she married as Jane, and once the other way round. Admitted one was in Kent and the other in Wiltshire, but it looks like a fairly common thing. Anyway, I did find this marriage on Ancestry:
Name: John Baynes
Marriage Date: 26 Aug 1793
Marriage Place: Saint Mary,Lancaster,Lancashire,England
Spouse: Jane Dawson
FHL Film Number: 0844800 IT 1, 0844800 IT
Actually has her down as Janes, but must be Jane. This would fit in with the earliest child's baptism in 1795.
I think this might well be a distinct possibility,