I have seen some terrible transcriptions and one yesterday was horrific. I was trying to find the 1851 census for Benjamin Read and his family. In the end I focussed on one of the sons, 'Walter', and then it came up in My Heritage with his name transcribed as 'MARY'. Alright it isn't very clear but when the very next entry reads 'Wife', surely whoever did this should have taken a second look?
I'm still not convinced about Jane. Like you Martin, I need something more substantial to link her to the Potter Family. In 1881 Edwin was a 'Contractor' and though I can't put my finger on it at the moment I think I did read a news report of him being an Engineer. However I have found in my own family and elsewhere that people quite often used the 'Engineer' occupation to make it appear that they had some higher qualification.
If Jane had just said 'Durham' I wouldn't have so much doubt, but she stressed it and even said 'City' which is a far cry from an outlying village. We have the very same puzzle in the Reed/Read search where the illegitimate girl is named Hannah Wilson sometimes and other times Hannah Ritchie. Her birth certificate reads birthplace 'Toft Hill', but in every census she says she was born at Cornsay which is far to the north of Toft Hill and near Tow Law and Consett.
In 1871 and in 1881 when Jane was 10 years old and then missing, the Potters lived in Middlesborough, so again far from Durham City.
Another thing that worries me is the age of Henry Thompson and where is he in any earlier census. We can be fairly sure that he did die in 1890, but there is no record of any Henry Thompson anywhere near that age. Then the Thompson family living right next door to Jane in 1891 - how were they related?