Ted, it is jurisdictional, but in SD County, I believe all small rooms (except off master suite) could be included in the bdrm count but there should be a notation in the comment section of the listing that one of the bedrooms is currently a study and does not have a built-in closet. It is simple to add a closet so it could be used as a bedroom.
Conversely, the assessor could state that a property has 3-4 bedrooms when it only currently has 2-3 because a former owner knocked down a wall between two bdrms to obtain a bigger bdrm. Lots of people do this without permits so the assessor would never learn of it. A new buyer is free to turn the big bdrm back into two bdrms.
I wouldn’t worry too much about the exact number of rooms the assessor states a property should have. A red flag on the assessment roll would be if it states the property is substantially smaller than what it currently appears to be (>=300 sf). This likely indicates a non-permitted room addition was built at some point in time for which the new buyer could have to get thoroughly inspected and pay back fees for (or possibly have to tear down) if he/she brings it to the attention of the planning dept while seeking permits to upgrade or change it.
Many houses built prior to 1933 do not have closets in all of the bdrms. In SF, only about 10% of all bedrooms there have built-in closets but yet they are ALL referred to as “bedrooms.” Ikea does a brisk business there selling various size “closets” (complete with drawers and organizers) :=]