Janet Booth's research is published in full below. She acknowledges the use of Parish Register Transcription CDs and Oxfordshire Marriage Index records courtesy of The Oxfordshire Family History Society.

"Going backwards from Zachariah GIBBS at Deddington 

17 December 1770
Zachary GIBBS & Amy GAYDON
10 December 1742 Mary daughter of Richard & Jane GIBBS baptised, buried 18 December 1742
8 January 1743/44 Zachariah son of Richard & Jane GIBBS baptised
9 August 1745 John son of Richard & Jane GIBBS baptised
19 May 1747 Mary daughter of Richard & Jane GIBBS baptised
30 November 1749 Amy daughter of Richard & Jane GIBS baptised
22 December 1751 Ann daughter of Richard & Jane GIBBS baptised
21 December 1753 Jane daughter of Richard & Jane GIBBS baptised
10 March 1756  
Phoebe daughter of Richard & Jane GIBBS baptised, buried 15 March 1756

It looks as if Jane GIBBS may have died as a result of childbirth because there is a burial for a Jane GIBBS on 13 March 1756.  Burial for a Richard GIBBS on 21 July 1790.  There is no marriage on the Oxfordshire Marriage Index for a Richard GIBBS (or variants) to a Jane.
 
15 November 1719 Zachariah son of Richard & Mary GIBBS baptised
10 June 1722 Richard son of Richard & Mary GIBBS baptised
28 February 1724/25 Susannah daughter of Richard & Mary GIBBS baptised
 
Burial on 29 April 1734 of Mary GIBBS, wife of Richard and one on 29 April 1746 for Richard GIBBS.
 
10 September 1682 Richard son of Richard & Mary GIBS baptised
17 September 1684

Ann daughter of Richard & Mary GIBS baptised, buried 23 September 1684,

  daughter of Richard GIBBS
18 July 1686
Ann daughter of Richard & Mary GIBBS baptised
15 December 1689 William son of Richard & Mary GIBS baptised
29 September 1695 Elizabeth daughter of Richard & Mary GIBS baptised
13 March 1697/98
Mary daughter of Richard & Mary GIBS baptised
8 June 1701 Alice daughter of Richard & Mary GIBS baptised

Burial 23 May 1693 of Mary wife of Richard GIBBS, so did Richard marry again?  Also a burial on 18 June 1700 of Jane wife of Richard GIBS.  So is she another wife of Richard above or the wife of Richard below and in that case the Richard & Jone GIBBS below are distinct from the Richard & Jane GIBBS parents of Richard baptised 1657!  Burial 9 January 1703/04 of a Richard GIBS.  Burial 17 January 1707/08 of another Richard GIBS.  This would make it likely that there were 2 distinct Richard GIBBs having children in Deddington between 1633 and 1660, one married to a Joan and the other to a Jane and that Richard & Jane were the parents of Richard GIBBS baptised in 1657.
28 April 1633
Thomas son of Richard GIBBS baptised
25 March 1638 John son of Richard GIBBS baptised, buried 1 January 1644/45, son of Richard & Jone

GIBS, a child
29 March 1640 Mary daughter of Richard & Jone GIBBS baptised
12 December 1642
Thomas son of Richard & Jone GIBBS baptised  
28 April 1644 William son of Richard GIBS baptised, buried 25 Dec 1644 infant son of Richard

GIBS the younger
24 August 1645 John son of Richard & Jane GIBS baptised
8 February 1645/46 Jane daughter of Richard & ? GIBS baptised, buried 12 May 1650 daughter of Richard
29 July 1649 William son of Richard & Jane GIBBES baptised
born 3 April 1652 Ann daughter of Richard & Jane GIBES
14 December 1654 Jane daughter of Richard & Jane GIBBES baptised
28 November 1657 Richard son of Richard & Jane GIBBES baptised
15 November 1660 Elizabeth daughter of Richard & Jane GIBBES baptised
 
Burial on 15 September 1663 of Richard GIBBS, a poor lab(ourer), followed by one in March 1674/75 of Joane GIBBS, a widow.
 
Deddington PR Transcripts only start in 1631, so I cannot go any further back for you, but the earliest GIBBS burial is one  on 16 August 1633 for an Anne wife of Richard GIBBS.
 
..... from this information it would seem likely that your Zachariah’s ancestors did originate from Deddington.   There is a baptism for a Richard son of John GIBBES at Hook Norton on 4 October 1640 but that is obviously too late to be the Richard GIBBS who is married to a Jane and having children in Deddington from at least 1645."