In Gee's book, she designated John Puckett, progenitor of the Henrico/Chesterfield Co, VA Puckett family as A1. John's three sons were designated as follows:
B1 - John Puckett born about 1654
B2 - William Puckett born about 1657
B3 - Thomas Puckett born about 1658
Gee recognized that both John Puckett (B1) and Thomas Puckett (B3) had sons named John, but missed the fact that William Puckett (B2) also had a son named John.
Henrico Co, VA OB 1719-1724:252 (007898203:142) - May 1723 - Last Will and Testament of William Pucket deceased presented by John Pucket his executor, proved by witnesses William Baugh, John Farguson & John Green. William Baugh & John Farguson security.
John Puckett who was executor of the will of William Puckett (B2) was almost certainly William's son. Note that the record book in which William's will was recorded is lost, we just have the court record where the will was proved and John Puckett affirmed as executor.
It certainly makes sense that all three sons of John Puckett (A1) would name a son John Puckett.
John Puckett (C15 in Gee's book) son of Thomas appears in this record: Henrico Co, VA OB 1719-1724:252 (007898203:142) - Nov 1723 - John Pucket son of Thomas for one wolf killed certified by Wm Kennon for 200 lbs tobacco [Note: the county government paid men in tobacco for killing wolves.]
John Puckett (B1), eldest son of John Puckett (A1), married later than his brothers. His will is found in Henrico Co, VA Miscellaneous Records, part 2, page 429 (LDS Film # 007898202, image 251/601). He names son John Puckett among other children; from the order in which his sons were listed in his will, it appears John Puckett III (C3 in Gee's book) was the eldest son. John Puckett III (C3) later administered the estate of his mother Elizabeth when she died in 1735.
John Puckett, son of William Puckett (B2) was the Executor of his father's will. John also appears to have been the executor of his brother Richard Puckett's (C8) will. Richard Puckett died at a time where we don't have regular court records or will records from Henrico Co, so we don't have his will, or even when it was proved in court. But in June 1720, William Randolph sued John Puckett as Executor of the will of Richard Puckett for a debt (OB 1719-1724:30 (007898203:24)). On 5 Oct 1725, The Henrico Orphans Court Ordered that churchwardens of Bristol Parish bind Richard Pucket orphan of Richard Pucket decd to John Pucket (Orphans:56 (007676115:251)). See page on "C8 - Richard Puckett" for John Puckett's records with Richard Puckett Sr (C8) and Richard Puckett Jr (D15).
Also, in Henrico Court, Sep 1721, William Randolph sued John Puckett as Executor of the will of William Puckett (OB 1719-1724:129 (007898203:80)).
6 Oct 1740 - John Pucket presents an account of the estate of Francis Westbrook decd. (OB 1737-1746:126, 007894525:70).