This mythical pedigree probably arises from confusion of the actual Earls of LennoX with the lairds/Barons of LennoCH.
The Earldom of LennoX came into the Stewart family in 1495, when the last of the original male line died, and various Stewarts and their descendants (the current Dukedom traces back to one of the natural sons of Charles II) have been Earls and Dukes of Lennox ever since.
The Drummonds of LennoCH never held a title higher than Baron, and tracing the earlier ancestors is not all that straightforward (even some of Burke's accounts start with the 8th holder, fl. 1640). There is no evidence that any member of this family ever emigrated to America. John Drummond, 8th of LennoCH, was certainly NOT the John Drummond who lived in Accomack/Northampton, VA and married Patience Hill, and neither was that John Drummond the son of the 8th of Lennoch (since the only son of record, the 9th of Lennoch, was named Adam).
According to Burke, the LennoCH Drummond line was a cadet branch of a cadet branch of the original Earls of LennoX; but the connection dates back to the early 15th century, or more likely the 14th century, and it is unclear whether the family had yet abandoned patronymics for inherited surnames (the LennoXes had not done so).
The Lennoch succession may be found in Burke's "Landed Gentry" under "Drummond of Megginch", but should be taken with a judicious amount of salt.
