"Half Monitor" is the roof.  "Clerestory" is the space inside the roof created by the windowed half monitor design.  Its a "churchy" term from the medieval period.  But both are used essentially interchangeably as the words become muddied by common usage.  Clerestory sounds better, I'll grant.  and your architect will understand it - but your roofer may not.
	
		
			
		
		
	
				
			
	