There definitely were Whiteheads on Long Island, and quite a few WikiTree profiles have been created for people with that name. Related families include Cornell and Skidmore. Inclusion of these people in the New Netherland project has been problematic for me. Their principal interaction with New Netherland seems to have been occupying territory (with the permission of the Dutch authorities); beyond that (and the occasional altercation with authorities) they had next to no interaction with the New Netherland community.
The most important recent publication about this family seems to be one by Harry Macy, Jr., (a respected New York genealogist): "The Family of Daniel Whitehead: A Century and a Half of Fact and Fiction." The New York Genealogical and Biographical Record 131.4 (2000): 263-75. From that publication, I get the impression that Daniel Whitehead was the first Whitehead on Long Island, and that he was the only Whitehead among the immigrant settlers. Daniel is one of the alleged sons connected to the John Whitehead profile you mention here, but there seems to be no indication that Daniel Whitehead of Long Island had a father living on Long Island. Moreover, a footnote on the first page of the cited article states:
Daniel's English origin is beyond the scope of this article. Different authors have suggested that he came from Dorset, Hampshire, Lancashire, London, Worcestershire, Yorkshire, etc., but no evidence of his birthplace or parentage has been discovered. The 1902 Cornell genealogy (note 23 below) stated that Daniel was "supposed to have emigrated to the Connecticut Colony with his brothers Samuel and Stephen who settled in New Haven" and this was copied by other writers but is totally unsubstantiated. Donald Lines Jacobus, Families of Ancient New Haven 7 (1931): 1970-71, shows Samuel Whitehead of New Haven with sons Samuel 1678 and Stephen 1680, and an Isaac Whitehead of New Haven who moved to Elizabeth, N.J. He makes no mention of Daniel Whitehead in this work.
That leads me to believe that Daniel Whitehead should be treated as "origins unknown."
I see that The Great Migration Begins (published several years before the Macy article appeared) repeated an indication from older publications that George and Thomas Alcock had a sister Elizabeth Whitehead of Leamington Priors, Warwick, England, who was the mother of sons named John Whitehead and Thomas Whitehead who had emigrated to New England. Macy addressed this information in another footnote, citing GMB and the 1997 Hatcher article (and earlier publications). His footnote says (in part):
Thomas Whitehead and his brother John were brought to New Haven as children in 1639, but there is no record of Thomas after 1647, while John married and left descendants. They were nephews of George Alcock. ... There is no evidence that these Whiteheads were related to those on Long Island. In all probability Thomas Whitehead died without issue.