Now that I have given you one hypothesis, let me do some work to give you a few of the alternative theories.
I hope you turn up something that no one else has found, that shows this link because the "official" take on Samuel Wright of East Haddam is that he just appeared there from nowhere and was not related to John Wright and no one knows where he came from. The fact that John Wright of Middle Haddam was not the executor of the estate of Samuel Wright of East Haddam, or in any way involved in the settlement of his estate is given as proof that they were not related even though they lived in close proximity to each other. All of that is certainly possible. There are over 100 completely separate lines of Wright men in America today whose most recent common ancestor predates the use of commoner surnames in England and Europe (ca. 1086 A.D.).
My observation is that it would not be the first pair of brothers that had no truck with each other later in life and this whole question of where Samuel Wright of East Haddam came from is not a simple matter by any means.
To give you the "official" theory (some of which I think is stretching it a bit): There is evidence that Martha Knowles was born 28 Jan 1713/14 in Eastham, Barnstable County, MA and was the dau. of Richard and Martha (Cobb) Knowles. A Martha Knowles married John Shaw (1700 - 1744) on 10 Nov 1729 in Eastham, Barnstable County, MA (at age 15? not terribly likely but not impossible either) and had children George (ca. 1730), Martha (ca. 1732) and John Jr. (ca. 1734).
John Shaw, Jr. moved to Wilbraham, MA before 1757 (before age 23) and then on to Cincinnati, OH in the early 1790's, while John Shaw Sr. and wife Mary Knowles moved to Hartford, CT where John Shaw Sr. died in early 1744. On 27 Dec 1744 Widow Mary Knowles Shaw married Samuel Wright in East Haddam and they had Jonathan (b. 13 March 1745, recorded in Hartford, CT), as well as Mary (b. 8 May 1748 in Litchfield, CT, m. Zachariah Spencer), Sarah (b. 5 Nov 1750, m. Thomas Purchase of Somers, CT on 31 Jan 1770 in Somers, CT), and finally, Samuel (b. 28 June 1753 in East Haddam, m. 1st Rebecca Dixon in Chatham, CT & 2nd Jerusha Bartlett in Chatham, CT). If this is the same Martha Knowles born in 1714, then she would have been 39 when her last child was born (an argument in favor of the early marriage to John Shaw). In any case, both Samuel and Martha died within days of each the New Years week following the birth of Samuel, leaving four orphans in need of foster care. Who took them in and raised them, I do not know, but it appears that they were not adopted, avoiding the complication of a change in their surnames. There should be some court records or something in the probate recoards of the county dealing with the need to find guardians for these four orphans of Samuel Wright and Martha Knowles. I have never looked for it.
NOW, let me spin a few more hypotheses for you to consider:
Samuel Wright, Jr. married both his wives in Chatham, CT, 1773 and 1796, respectively. Juxtaposition this information against the fact that the grandson of John Wright "of Barnstable", Capt Noah Wright, was born in Chatham 1771, which is to say that the son of John Wright "of Barnstable" and Middle Haddam, William and his wife Ann, lived in Chatham during the time that Samuel Wright, Jr. was getting married there. Coincidence or kinship?
NOW for a third thread to consider. John "of Barnstable" may not refer to the town of Barnstable, but may, instead, refer to the MA County in general. If that is the case, John could have been from Sandwich or a number of other places in Barnstable County, MA. If we theorize he was, for instance, from Sandwich, Barnstable County, MA, he would most likely be descended from Peter Wright of Long Island whose son, Peter, Jr. was born in Sandwich in Mar, 1636/37. These people were Quakers and were not well received by the Puritan founders of the town so that eventually the descendants of Peter Wright moved to other places where they would be more accepted, or repudiated their Quaker faith. The list of male descenants of Peter Wright have not been fully researched. There is considerable question about who many of the third and fourth generation were. Your Samuel could be one of them as well.
As for the Plymouth/Plympton hypothesis, there is also a Y-DNA profile for the Long Island Wright group of brothers Peter, Nicholas and Anthony Wright, who are descended from the Kilverstone Hall Wright's of Norfolk, England.
I have names and dates for a great many of the descedndants of Samuel Wright of East Haddam up to about 1850. Beyond that, I have not traced any lineages down to the present day, and, of course, I have no proof of any parentage for Samuel, one way or the other. These are all proposed working theories which beg to be researched to eliminate or confirm with documentation.
I also have considerable information on the Oyster Bay Wrights of Long Island, if that would be of any help to you.
I think it is time to end this and the confusion it unavoidabley brings. Sorry, but this one has got at least four working hypothesis that are wide open for more research, so far as I know.
I will help you get started with what data I have, if you need it. Please let me know what you want to tackle first and I will send you all I have on that line.