But you cannot designate what a room is to be used for, only the HO can say how the room will be used and if the HO says it's a closet then no recept is needed.
I disagree. Lots of times HO or EC will try to do exactly what you are saying and call a room something else just to get out of doing something that they should be doing in the first place. Which is a piss poor practice IMO. The inspector has the right to cut through the BS and call it what it is.
For example around here to get out of AFCI protection people will say "No this is not a bedroom it is a bonus room it just happens to have a closet and a bed in it."
If you want to call a big room with built in shevles along the wall that is going to be full of books a closet that is fine but I am going to call it a library and install receptacles in the toe kick of the shevles so they will have somewhere to plug in some lamps or anything else they might want.