I don't know where Dr. Williams get's his definition but every on-line dictionary I've checked defines a civil war as " war between opposing groups of citizens of the same country." Which is an accurate defintion of what happened.
Whether a war is later labelled by historians as a "Civil War" or a "War of Independence" is determined by who wins.
Before the defeat of the British at Yorktwon, the British considered the Americans as citizens of the same conutry in rebellion. Was it then a "Civil War" or a "War of Independence"? Both sides agree it was a "Revolution".
During the 1861-1865 period, contempory writings on the Confederate side describe the war as a "War of Independence" and contemporary Union writing always refer to a "Rebellion".
The term "Civil War" came into common use in the North at a later date as a less "in your face" term to use than "The Great Rebellion".
If the Confederate States had won, it would have been called the "War of Independence".
Yeah but you didn't win. So a rebellion it was, and a rebellion it remains.