Research published in 2004 identifies the father of the immigrants William and John Potter as William (not John) Potter.
John Potter, son of William, was baptized 18 Feb 1609[/10], in St. Thomas-in-the-Cliffs, Lewes, Sussex, England.; m. say 1635, prob. in England Elizabeth _____; went to New England
John Potter married Elizabeth UNKNOWN about 1635 in England. Elizabeth died 28 July 1677 in New Haven; she married second Edward Parker and m. 3rd Robert Rose.
John's brother, William, and his family left England on the Abigail John was not on the list and followed later.
He was in New Haven Colony in 1639 when he signed the Fundamental agreement.
John died in 1643. ? 10 Aug 1644 ?. This date is incorrect. John Potter did not sign the Oath of Fidelity given 1 July 1644.
John b. c. 1636; d. 24 Dec 1706; m. (1) Hannah Cooper; m. (2) Mary Hitchcock, widow of Ralph Russell
Hannah d. 7 Nov 1723; m. (1)Samuel Blakeslee; m. (2) Henry Brooks
Samuel bp 7 Oct 1641 New Haven; d. 1696 m. (1) Anna Russell; m. (2) ____ ____
N1. John was NOT the John Potter who married Elizabeth Wood in Chesham Co., Bucks, England on April 14, 1630. Jacobus asked Was he ... 
↑ Hatcher, Patricia Law. "English Origin of the Potter and Beecher Families of New Haven, Connecticut," in The American Genealogist 79(2004):28, citing St. Thomas parish records, and register of wills, Lewes, Chichester. (Online database AmericanAncestors.org. New England Historic Genealogical Society, 2009 - .) Volume 79, Pages 28-32.
↑ "England Births and Christenings, 1538-1975," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:NDZV-1ZZ : 10 February 2018, John Potter, ); citing St. Thomas-in-the-Cliffs, Lewes, Sussex, England, index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 1,067,241. image
↑ 3.03.13.23.33.43.5 Jacobus, Donald Lines (compiler). Families of Ancient New Haven, Vol I-VIII. and Index Vol IX New Haven: 1931. Reprint, Baltimore: Genealogical Publishing Co., Inc., 1974, 1981, 1997. Originally published as New Haven Genealogical Magazine, Volumes I-VIII. Rome, NY and New Haven, CT 1922-1932.
↑ "William Potter" Great Migration 1634-1635, M-P. (Online database. AmericanAncestors.org. New England Historic Genealogical Society, 2010.) Originally published as: The Great Migration, Immigrants to New England, 1634-1635, Volume V, M-P, by Robert Charles Anderson. Boston: New England Historic Genealogical Society, 2007. https://www.americanancestors.org/DB401/i/12155/512/235143778 p. 512
You say that this is not the John who married E Wood. Where does this claim come from? Is that John on Wikitree?
I will add a source that contradicts this, though it is not necessarily reliable. This source is a DNA match to me up to John (1607). It doesn't require that my John be the son of William, but that is one possibility.
Post merge message: I think we've merged the duplicate Johns; before we merge the duplicate spouses, can someone please summarize what is documented about them? Are these the right spouses? Which are the correct maiden names for them? Thanks!