This is not the Peter from Warrington. That Peter was a tobacconist and died there in 1773.
Baptism records for "Deane by Bolton" and "son of John" include Peter (1734), Catherine (1737), Thomas (1740), Samuel (1744), Thomas (1745), John (1746), John (1747), Ann (1748) , Richard (1750), Betty (1751), John (1754), Mary (1755), and so on. There must be at least two families here, but also indicate that this a Deane family (who had a Richard born 1750), cannot be the same as the Turton (Walmsley) family with brothers Peter and Richard (and possibly John & James).
The family have been shown not to be the ones of Middle Hulton, but possibly have been in Turton (or Rivington or Haslingden) for several generations. Turton Baptism records are lost for this period.