From the comment trail it appears that we need to clarify who some of the players in this drama are. In American colonial history the Plymouth Colony settlers were a mixture of Church of England and Separatist adherents. Today many British writers call Separatists ‘Independents,’ euphemistically meant to soften the persecution they experienced at the hands of the established church. Separatists are an English phenomenon. Many of them settled in Leiden. They believed that the established church was so corrupted with Catholic dogma and practice that it was irreformable. The only way to sound, uncorrupted worship was through separation. The crown and church saw this as treason and persecuted them mercilessly.
While Lutherans, Calvinists, Anabaptists and others separated from the Catholic Church and were – like Separatists – Protestants, the term Separatist applies ONLY to the English phenomenon. Other than English exiles living in the Netherlands, there were no European Separatists.
Puritans were also a uniquely English growth. While there were those in Europe who sought pure doctrine and practice, Puritanism refers to those who wished to reform the English Church. Unlike their Separatist brethren, they believed the English church was reformable. They sought reform through political power; the result was the English Civil War and abuses as sever as any under the king and church.
These are basics of American history because much of this story is the founding narrative for colonial era history. But surely at least some of this is taught in UK schools. Perhaps not. Each country’s textbooks foster myth. Myth is as surely created by omission as by falsehood.