Pilgrims and Puritans are generally thought of as being the same people.  They definitely were not. They didn't even like each another.  The Pilgrims are the ones that arrived on the Mayflower in 1620 at Plymouth.  They were not of Puritan Religious belief (Pilgrims wanted to be separated from the Church of England while Puritans wanted to remain as members of the Church of England, but wanted it to return to a more "pure" form).

And it is the Pilgrims who celebrated the first Thanksgiving, not the Puritans.

So here we may have a potential means of separating some of our different John Carman's according to when they arrived and whether they are the same men moved to different locations at different times.

The John Carman's who arrived in this time period are John Carman of the "Fortune" and John Carman of the "Plough".

The Puritans did not arrive until 1630 and their founding Colony was at Salem. (there are Puritans that arrived before 1630, but the vast majority began to arrive then and started their Colony).

Puritans believed that religion was an intellectual pursuit and differed from most establishments in that they believed that every man had the right to educated and engage in intelligent discussion on the scriptures.  Harvard University was born and survived because of the Puritans who believed every man had the right to and should be educated to better understand the revealed scriptures, not just the aristocracy or Church bureaucracy.  This also lead to the establishment of the first 'public school' in Boston in 1635.

The John Carman who arrives in this time period is John Carman of the "Lyon".

It should also be noted that the Carman family that I am interested in were members of St George's Church in Hempstead, Long Island when it was finally built - it was later and Episcopal Church but foremost it was established as a  Church of England and one attended even by the British Army during the American Revolution.

And going in the opposite direction in time, it was William and Thomas Carman, Puritan Martyrs, who were burned at the Stake in the sixteenth century.