Were the Pilgrims, pilgrims amongst themselves? Perhaps they called themselves Puritans? …did they see themselves on a pilgrimage? - Jinx
I believe the saw themselves as seeking religious assylum and a better life in the new world. I give them much credit for actually making it…Life was extremely hard for them. I live not too far from plymouth and right next to a settlement built in the 16th century. Think large palicades around homes and literal cutting a living out of the forest.
I’ve been trying to find contemporary accounts online. This is apparently the only account of the journey of the Mayflower to be written by someone on the boat, William Bradford:
http://www.mayflowersteps.co.uk/mayflower/journey.html
It calls the travellers “passengers”. The same page has an account by Captain John Smith of Pocahontas fame, who uses similarly non-specific terms. The compact they signed before the voyage sets out their intention as colonists but never names them as a group. There must be other letters referring to the voyage, if somebody else can find them.
35 of the 102 of the passengers were members of the English Separatist Church, or Separatists for short, according to Britannica. But that only applies to a minority.
Most of the people on the Mayflower were going there as settlers or colonists, as Phlosphr says. It’s possible they didn’t need a specific name for themselves beyond that.
Well…
I went next door, and posed the question to the head of our History Department. His reply was that “they called themselves Pilgrims.” He also noted that in Winthrop’s “City on a Hill” sermon, he makes constant reference to “The Chosen Ones.” (Who knew the Pilgrims were Jewish?) 
What Did the Pilgrims Call Themselves?
John Wayne. 
The Pilgrims did not call themselves Puritans, as The Pilgrims and The Puritans were two different sets of people.
The Pilgrims were members of a religious sect known as Separatists. They saw themselves as a distinct denomination. As noted above, the custom of referring to them as “pilgrims” apparently derives from a remark by William Bradford.
It was The Pilgrims who arrived at Plymouth, and it is in imitation of them that Thanksgiving was created.
The Puritans were the founders of the Massachusetts Bay Colony, a different settlement than the one at Plymouth. They are said to have not regarded themselves as a separate denominaton from The Church of England. Rather, they were reformers who held to H. L. Mencken’s famous definition of The Puritan Ethic; they had a suspicion that other people were having fun.
And they were pretty much opposed to that. The Puritans wore dour, dark clothing which amounted to uniforms. I recall reading once that only one contemporary portrait is extant of a Pilgrim, and he was decked out pretty much like a pimp, with a ring on every finger.
The pictures which are used for Thanksgiving decorations, then, are of people who had nothing in particular to do with the first Thanksgiving.
Incidentally, I recall that when I took Civics class in high school the textbook quoted Lenin as saying that “they do who do not work, neither shall they eat”. The implication was that this proved that Lenin was a perverse monster.
When The New Yorker published a Bicentennial issue in July of 1976, the cover illustration was a clever drawing of Uncle Sam which was made up of famous quotations by famous Americans. The topmost one was credited to William Bradford: “they who do not work, neither shall they eat”. The implication here was that this was a hearty expression of American grit and practicality.
It seems likely both men were aware that they were quoting St. Paul.
There were no groups in the 1620s who called themselves ‘Separatists’. The word did exist (although it was a relatively new coinage), but it was only ever applied by other people. Historians now use it in connection with the Plymouth colonists only as a knowing anachronism in the absence of any convenient contemporary term.
This site has this brief discussion as to what ‘the Pilgrims’ actually called themselves.
Not everyone along for the boat ride was a “Pilgrim,” either. IIRC, only about half of them were. The rest were regular folks seeking to make their fortune in the New World.