In my french books on Army of Italy, the piemontese militia/partisans against the republican french are always called "Barbets", not Barbetti.
Also, I spent years wandering the Alps, including the Vaudois villages, and I never read or heard the Vaudois to be named Barbets.
Could you please provide your source ?
Clearly, that is a French equivalent, in a book written from the French point of view, of the Italian/Piedmontese term 'barbetti' which in the context is obviously the more accurate term of reference; in the same way that most people would, I am sure, be more inclined to write Garde Imériale rather than, say, Guardia Imperiale.
If you enter barbetti & Piedmont in your chosen search engine, as I did, you ought to have an instructive couple of hours. Similarly, enter barbet and Waldensian (etc).
A good starting point is Michael Broers- 'Revolt and repression in Napoleonic Italy 1794-1814' in War in time of Revolution (2012)
More general is his Napoleon's other War- bandits, rebels and their pursuers in the Age of Revolutions