That’s a great find by mckha489, and confirms your story about the child George and Jane’s death.
Jane’s death certificate could be useful, even if parents’ name are not there, knowing the address and name of informant might be important to moving forward.
I wonder if George’s birth certificate, rather than death, might provide more useful information – here again looking to see what the address at birth was, etc.
George Dickson’s birth certificate ref no is 691 / 264 Lasswade – 1875.
The 1875 valuation rolls for Pentland Villa, St Cuthberts show Mrs Mary Duncan (Proprietor/ occupier), Peter Howden (proprietor), and Henry H Hanson (Tenant).
I think that the census record you found for Jane in 1851 and 1861 was for the Jane who married Hugh McInnes. Daughter of William and Catherine, born in Roslin.
1851 census here -
https://www.freecen.org.uk/search_records/5903d689e9379091b1f69984/jane-dixon-1851-midlothian-lasswade-1846-?locale=enI would also look at the details of William Dixon’s birth certificate in 1863, and see what the address is.
What became of William, and where is he in 1881 census?
If you have a DNA link to this family, then I would look further into this family – siblings of William Dickson, as a potential parent for Jane.