This answer relates to your post with the info and citations you wanted to add:
Perhaps I am missing something, but I am not sure how a William Gibbson who died in 1684 in London, and who was a supposed son of Sir William Gibson relate to the immigrant John Gibson? Looking at Sir William's FS profile, though, it does list a supposed son, John, born in 1601. One of the sources is Anderson's ''Great Migration'' which echoes the approximate birth year of 1601, but it says his origins and parentage are unknown.
Of the "sources" shown on John Gibson's FS page, there are no references there to his father being William, although he is connected (without sources) as a son to Sir William Gibson there. I see one reference on FS showing his father may have been Alexander (discussed on John's profile).
The obituary is not for this John.
Thanks for posting this question, and good luck in your search. It is frustrating to find brick walls like John Gibson's...