Pics of the ESU trucks. Their website also references "sworn" officers, so I'd say the "Department of Public Safety" is a kinder, gentler way of saying police, although they are probably also responsible for the fire alarms, sprnikler system maintenance and other things that a "normal" police department would let a fire depatment or EMS agency deal with.
I believe the special patrolman status is correct. It is a private agency and the authority of their members essentially ends once their are off duty. They do not carry off duty on their badge or anything like you typically see from other govt law enforcement agencies. They would need to apply for a pistol permit if they wanted a gun off duty and they are restricted to the arrest powers of your average Joe citizen once they are off the clock. The agency would most likely not offer indemnification for off duty incidents and I would have to believe it probably goes against a policy. I believe their "ESU" duties are reminiscint of what Housing Emergency Service did decades ago like responding to stuck elevators and performing rescues in that sense or people trapped on scaffolding etc. Their response is heavily tailored for ems assists and they are not going to be the lead agency responding with heavy weapons on a barricared EDP's etc. That said, it is probably a good experience for someone who lives in NYC and is testing the waters thinking about police work. They are recognized as peace officers in the Criminal Proceedure Law, not Police Officers despite the Police title