Both Stone and Stow-in-the–Wold baptismal registers are covered by the IGI for the period of interest circa 1780.
Stow 1610-1812
Stone 1578-1812
There are Betteridge baptisms showing on the IGI in Stow beginning in 1785 for John and Elizabeth Betteridge and 1788 for Robert and Elizabeth Betteridge. I can’t find marriages on the IGI for either of these couples, but then Stow marriages are not covered by the IGI. Neither have a son called John on the IGI baptised in Stow. Though a Mormon church member submission 'guesses' at a birth in Stow for a John Betteridge.
John Betteridge
Birth: 1785 Of, Stow On The Wold, Gloucester
Father: Robert Betteridge
Mother: Elizabeth
Spouse: Mary Cary
Marriage: 18 APR 1809 (according to an extracted entry in Wotton Under Edge)
There are no Betteridge baptisms in Stone.
On the 1851 census there are Betteridges living in Stow who were born there about the same time as John in Eastbourne. William Bettridge aged 60 and Thomas Betridge aged 63. There are other Betteridges born Stow.
There are no Betteridges born Stone from any period.
I have access to a burial index recording burials in Stone and Stow from 1813 onwards. No Betteridges in Stone but quite a few in Stow. For instance
1823 John aged 63 birth year circa 1760
1829 Elizabeth aged 69 birth year circa 1760
1833 Elizabeth aged 80 birth year circa 1753
1851 Robert aged 83 birth year circa 1768
If John’s age on the 1851 census in Eastbourne is correct, 71 (and his age at burial or on his death registration would help confirm it) then with a birth year of circa 1780 John of Stow or either of the two Elizabeths would be old enough to be his parents.
Gloucestershire Record Office online database shows no instances of Betteridges in Stone but it does show some interesting records for Stow.
http://www.gloucestershire.gov.uk/index.cfm?ArticleID=1335Year Surname Forename Gender Parish Reference Occupation Catalog
1823 bettridge john U stow-on-the-wold 1823/ 19 shopkeeper Wills 1801 - 1858
and an earlier will
1808 bettridge robert U stow-on-the-wold 1808/ 80 carpenter Wills 1801 - 1858
There is also a Prerogative Court of Canterbury will held by The National Archives for Bettridge in Stow but again nothing in Stone.
http://www.nationalarchives.gov.uk/documentsonline/Will of Elizabeth Bettridge, Widow of Stow on the Wold , Gloucestershire 24 October 1829 PROB 11/1761
and the Death Duty entry for Robert’s will
Description Administration letter of Robert Betteridge of Stow, Gloucestershire. Proved in the Court of Gloucester.
Date June 04 1808
Catalogue reference IR 26/346
Dept Records of the Boards of Stamps, Taxes, Excise, Stamps and Taxes, and Inland Revenue
Series Board of Inland Revenue and predecessors: Estate Duty Office and predecessors: Registers of Legacy Duty, Succession Duty and Estate Duty
The National Archives research guide gives details on the sort of information that might be found in Death Duty registers.
Death Duty Records, From 1796
http://www.nationalarchives.gov.uk/catalogue/RdLeaflet.asp?sLeafletID=107‘They can give the name of the deceased, with address and last occupation, the date of the will, the place and date of probate, the names, addresses and occupations of the executors, and details of estates, legacies, trustees, legatees, annuities and the duty paid. They can also give the date of death, and information about the people who received bequests (beneficiaries), or who were the next-of-kin, such as exact relationship to the deceased. Tax was not payable on bequests to people within a closely defined family circle, and as a result the family relationship was often noted in the registers.’There are no wills at TNA at Stone for Betteridges.
None of this is conclusive for a birth for John Betteridge of Eastbourne in Stow (as Andrea says the census enumerator's handwriting on the 1851 census isn't easy to decipher), but of the two places, Stow or Stone, it certainly seems to prioritise the search to the possible Stow records first.
Regards
Valda
