This summary is not the rule. The rule text controls.
This rule explains who counts as a "public administrator" under New York law. A public administrator is a government official who handles the estates of people who die without family members available to manage their affairs, or without a will naming someone to do the job.
The rule says that whenever this term is used in this section of the law, it refers specifically to the public administrators in six counties: Erie, Monroe, Nassau, Onondaga, Suffolk, and Westchester. This matters because these counties have their own public administrator offices, while other counties in New York may handle these duties differently, often through the county treasurer or another local official.
If you live in one of these six counties and a relative dies without a will and without other family able to step in, the public administrator's office in that county may end up managing the estate. Knowing which counties this rule applies to helps people understand who to contact if they run into this kind of situation.