What do you mean by Puritan? The Baptists are the heirs of the Puritans.
Yes, we are. ;-)
Which Baptists?
Never seen two Baptists agree on more than 3 things, and I are one.
Perhaps the Reformed Baptists, who are Calvinist. The Congregational Church, now known as the United Church of Christ (UCC) grew out of Puritanism, but it is today one of the pillars of the Religious Left. Barack Obama worshipped at a UCC church headed by a radical pastor.
Oh no, Baptist consider themselves heirs to people like the waldenses or albigenses. As far as European sects that are Christian but non-calvinist, I’m not quite sure but the Puritans actually outlawed Rodger Williams and banished him to the area that we now know as Rhode Island because the Puritans of Plymouth Colony didn’t want Baptist there