"John Baker became a resident of Hartford, Connecticut in 1665, settling on Baker Lane (now Ward Stree)....His origin remains unknown, although he is believed to be of English extraction...."
Church records for the pre-1685 period have been lost.
After the death of wife Lydia, John married Rachel (Ballard) Merry (widow of Cornelius Merry). (No children) [Note some authors incorrectly have this marriage to John Jr, and incorrectly report the date)
Born 1640 or earlier
Death: Hartford, CT - after 27 March 1704 Citing a deed to son William on that date (Hartford Deeds 1:29)
Sources
Baker, Fred G., "The Descendants of John Baker (ca. 1640-1704) of Hartford, Connecticut, through Three Generations", Connecticut Nutmegger (Connecticut Society of Genealogists, Glastonbury, Conn., 2011) Vol. 44, Page 194.
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I added some notes from the Connecticut Nutmegger article, which is pretty thorough. There doesn't seem to be anything to connect John Baker of Hartford (h. of Lydia Baysey) with Boston. I suggest that his birth date be updated to an estimated 1640 (place be left blank), and his death date be changed to after 27 March 1704 (Hartford, Connecticut (uncertain)....and definitely be detached from mother Hannah, who is too young to be his mother anyway.
Baker-4803 and Baker-3948 do not represent the same person because: Baker-4803 lived in Boston and Baker-3948 lived in Hartford. Birth date of 20 Jun 1640 is correct for Boston John Baker. Not certain it's correct for Hartford John.