Let me first say I am a Bynum reseracher and know little about Abernathys.However, I think your suspicion that your Robert H is not the same person as the son of Robert Abernathy and Dicy Bynum may be correct.
A Robert Daniel Abernathy who died in 1900 left behind a manuscript with some information re his family.He stated that his grandfather was Robert Abernathy and grandmother was Dicy Bynum.He lists their children as "five sons, no daughters" named Miles Osburn Abernathy, Robert Hager Abernathy, Samuel Abernathy, Daniel Abernathy, and Rufus T. Abernathy ("the youngest").He also says his grandfather Robert Abernathy "lived and died" in Lincoln County.
R D Abernathy also gave his grandfather's brothers as Miles, William, and Shade (omitting for some reason Joseph).
The "Abernathy Information" of Margaret Butler includes several letters written 1828-1841 by Robert Abernathy (the grandfather) to his brother Joseph Abernnathy in Cape Girardeau, and later Jackson, Missouri.These speak of their brothers and a number of other family members scattered across several states.
One of them dated 22 March 1840 speaks of the death of "Robert Martin brother-in-law to [my] son Robert."
Also, in a letter of 13 January 1828 Robert Hager Abernathy, son of the above Robert, writes that "grandfather is tolerable well... father is teaching school near home..." and he says "I am now living at the Catawba Springs near Beaties Ford with Mr. Charles Jugnot, lately from France, but if I do not live here, I expect to live with John Hays."
Perhaps these clues would help resolve the identity issue?
By the way, Robert Abernathy Sr.'s letter of 25 March 1832 to his brother Joseph Abernathy reports the death of their "ancient and beloved" father (unnamed) on "Thursday last".He estimates his father's age at 95.