Eric,
I've just scanned the British Library for that name between 1791 and 1841, and the only five English entries are for winners of fruit and vegeatble competitions in Birmingham in 1836 & 1837. Unfortunately, they just give Peter Gibson as a winner in particular classes and no other details.
However, it's also interesting to note that over 150 entries out of a total of around 280 were actually in Ireland. So that is another possibility. Having said that, the names of his children (Maria, Peter, Thomas, Henry, Richard & Arthur) don't include any any of the usual irish names such as Patrick or Mary.
Looking for possible Gibson siblings born between 1786 & 1796 in the Birmingham area, there's only 7 possibles, most born in Birmingham or Aston, one in Notinghamshire and two in Scotland. There's a Richard in Aston amongst them born in Warwickshire at what looks to be Stamber Mile, but there is no such place in the UK, let alone Warwickshire. See what you make of it:-
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Maurice