We need to be very careful about the Thayers. Research published in 1906 and subsequently for many decades relied on what turned out to be inaccurate transcriptions of the Thornbury parish records-- and even some old wills there. Therefore, information citing that old 1906 and subsequent articles which also rely on these bad transcriptions, so anything on wikitree using that source needs to be double-checked.
I added this to the immigrant's profile:
Clifford Stott published in 1998 a serious and critical review of past research done on the Thayer family, and pointed out MANY errors of previous transcriptions. He re-analyzed the original parish records of Thornsbury, and published a new, corrected transcription of them. He also transcribes early probate records.
Source: Clifford L. Stott, "The Gloucester T(h)ayer Ancestry," in TAG,73(1998):82-86
There's also a 1995(?) genealogy of the Thayer family, but it's not online; no hard copy at Library of Congress, but NEHGS in Boston has a copy.