I hear you, but I do not agree. Puritan is the catchall term used both in the period and today to refer to the group as a whole. Puritans as in anti–Church of England, Protestant radicals, Pilgrims, the Mayflower, all that stuff. Shakespeare uses it, and I’m rather betting that more of us will have encountered Malvolio in the original than Edward Taylor or Cotton Mather or Roger Williams or Jonathan Edwards or Anne Bradstreet or Samuel Sewall und so weiter. So, yeah, Puritan.
There are indeed other terms: Brownist, for example, or Separatist, or, perhaps the best, Congregationalist. One sees these in the texts. But when I teach this material or talk about it, Puritan is the one term I can count on everyone knowing, regardless of how, say, it lumps together someone like Anne Hutchinson with, say, William Bradford, two very dissimilar Puritans. They liked using the term saints, but try explaining this term to an audience used to thinking of Catholic saints (or the figurative senses derived therefrom).
Jamestown was indeed the first recorded ceremony of thanks in America: but it’s the Plymouth version that gave us the holiday and that’s come down into popular memory, so I dunno about that. And the Puritans were incredibly strict on a number of things: sexual mores, land rights (they hated the “adventurers” who came over with them but didn’t identify as Congregationalists), dealings with the Native Americans (although here, indeed, even a cursory read through the texts shows them to not have been the two-dimensional Hateful Racist European Colonizers of popular legend), oh, yes, and religion. So again, I dunno.
The word works because it’s a) accurate (it refers to the people we’re talking about) and b) it refers to exactly the tiny group of colonists that it’s intended to refer to: the Puritans.
My social psych theory holds plenty of water, being largely cribbed from Perry Miller and a legion of Americanists. And the viability of the term, its cultural “legs,” if you will, has little to do with the accuracy of the term when applied to its original context. The 1950s may or not have been Puritanical (ask Hoover for a fun response!); the 1650s, um, yeah, Puritanical.