I corresponded a few years ago with the wife of one of your Gordon cousins. As both are still living (and hopefully for many years) I won't mention names on the board, but she also told me that she could not get her husband's aunts to discuss the issue.
It is amazing the lengths to which people would go to avoid the issue of illegitimacy. Although Sam's birth was registered at the time as Samuel Watt Jeffrey, it seems that when he needed proof of his age many years later he did not ask the Department of Vital Statistics to search the records under that name. Instead, On April 2, 1934 his aunt, Margaret (Jeffrey) Olmstead declared before a Notary Public at Benito, Manitoba that Samuel Gordon was born in Ross Twp. on January 10, 1871, that the father was Hugh Gordon and the mother was Mary Jeffrey, and that she was present when the child was delivered by Mrs Gilbert Edmunds.
Of course it is always possible that he knew of the earlier registration, but that it was not acceptable for whatever purpose he needed it because the name did not match the name by which he was then known. I expect that his adoption by Hugh Gordon was a de facto adoption and that there was no documentary evidence to support the change of name from Samuel Watt Jeffrey to Samuel J. Gordon. Since by 1934 both Hugh and Mary had already died, this little white lie may have been seen as the only way to resolve the matter.