What religion is Kit Tyler?

Kit Tyler is an Anglican, a member of the established Church of England. This is just one of the many characteristics that sets her apart from the Puritans of Connecticut. The Puritans were a group of English ultra-Protestants who believed that the Reformation hadn't gone far enough. They wanted to purify the Church of England of all remaining traces of Catholicism. (This is how they got the name Puritan).
Although they were in the ascendancy during and after the English Civil War, they were subject to official persecution and intolerance when the monarchy was restored under Charles II in 1660. It was during this time that many Puritans fled to North America to practice their religion in peace.
It's not surprising, then, that the locals should be so suspicious of Kit when she arrives among them. She's not a Puritan; she's a member of a Church that the locals regard as impure and ungodly and whose governors, along with the state authorities, were responsible for driving them out of England.