The closet 'defines' the room as a bedroom, so it's probably more related to marketing the home as having "x" number of bedrooms. In older homes, as you know, there were no closets, so this general rule would not apply. Bedrooms must have a minimum of 2 exits and provide adequate ventilation, plus be a minimum of 7'x7' with 7.5' headroom. Local building codes would clarify this in the area of the dwelling. A licensed appraiser can address this issue from their perspective.
Putting myself in the shoes of a homebuyer when considering the purchase of a home , most if not all will call the space a Den or Office if there is not a closet. Simply put, if they have 3 kids and they all want separate bedrooms and the listing agent is marketing it as a 4 bdrm home...someone's going to be without a closet, and the buyer's will tell you, 'it's NOT a bedroom.' I have run into this time and time again. As a Broker I always check with public record to see what the County shows as # of bdrms. ...Hope this helps Joe.
To an appraiser . . . well, it depends. As a broker, I just go with what is customary; some 50's houses have bedrooms with small windows high up on the wall; some '20s and '30s buildings don't have closets in the bedrooms. It has two bedrooms without closets - is it really a zero bedroom, 1.75 bath co-op? No!