The Annals of Newtown by James Riker, Jr., published in 1852, states, “The earliest record of this family in America is in 1637, when John Burroughs is found in Salem. He soon after came to Newtown. He filled the office of town clerk several years. His sons Jeremiah and Joseph shared their father’s estate. Jeremiah died in1698.”
This matches John, son of Jeremiah, son of William, Vice Admiral. If the family arrived on a companion ship of the Hector in 1637, stopped first in Salem, MA then to New Haven in 1638, Riker could safely refer to him as John Burroughs "of Salem."
When writing of the re-settlement of Mespat, present day Queens, Riker says,
“But in 1652 a goodly company of Englishmen arrived from New England. Some of the new corners were from Greenwich, Stamford, Fairfield and villages along the Connecticut shore,,,, Of these were ... John Burroughs from Salem, ...
We know John Burroughs of Newtown sold his mother in law's property in Stamford CT in 1652,
IMHO, John of Salem is John of New Haven, John of Stamford and John of Newtown.