The short answer is that both are correct.
This article was first published in BBC History Magazine in 2012
Thursday 1st December 2016
The group of English colonists who settled in North America and later became known as the Pilgrim Fathers originated as a group of Puritans from Nottinghamshire and Lincolnshire who by 1605 had come to believe that their Christian faith was incompatible with the Church of England.
The group decided to leave for the Netherlands where they expected to be able to worship freely and in 1607 a ship was hired at Boston to carry the 150 members of the congregation to Leiden. Unfortunately the group had failed to get the necessary passports and were arrested when they tried to board the ship. A monument stands on this spot to mark the event.
The following year the group did emigrate to the Netherlands. Then in 1617 they heard of an offer of free land in the new English colony of Virginia, subject to various tenancy agreements. After some protracted negotiations with the colony’s administrators, the group decided to leave for the New World.
A ship, the Mayflower, was hired to carry them to America and in July 1620 the group set off. The ship put in at Plymouth to await favourable winds and pick up supplies. They finally left in September 1620 to sail into history.
Answered by Rupert Matthews, historian and author.