Might have found out a little on Richard BAKER, father of John Thomas Wright BAKER. In 1851 he was listed as being a widower and his age suggests a birth of about 1774. His place of birth was listed as Cawston, Norfolk, England in the 1851 census record. In the 1851 record he was listed as being the Rector of Rotley MA. There was a christening record for a Richard BAKER who was Christened on 1 October 1773 and listed as the son of Richard and Elizabeth BAKER. The christening record - "England Births and Christenings, 1538-1975," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:NNDW-1W5 : 12 February 2018, Richard Baker, ); citing Item 42, index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 1,657,158. - stated on the record that he was born on 28 September 1773. The bottom of the page was signed, Richard BAKER [Rector].
Now although I have not found the proof of it, it is possible that the Richard BAKER who was born on 28 September 1773 in Cawston, Norfolk, England to Richard and Elizabeth BAKER was the same individual as Richard BAKER, the Rector of Rotley MA, and father of John Thomas Wright BAKER. That would be two more generations back if this connection could be proven.
- Possible 1841 - "England and Wales Census, 1841," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:M79Z-WZR : 12 December 2017), Richard Baker, Botley, South East Part, Hampshire, England; from "1841 England, Scotland and Wales census," database and images, findmypast(http://www.findmypast.com : n.d.); citing PRO HO 107, The National Archives, Kew, Surrey.