The "OR" that you highlighted is between "POLICE OFFICER" and "PEACE OFFICER." It has nothing to do with the actions, but is instead making the actions a crime if the target is either a police officer or a peace officer. I have no idea what the difference is between those two in NY.
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There is slight differences in Authority. It actually is quite complex from reading the NY Penal law and seems to be more a function of the agency then of power...
However from reading the appropriate laws it appears to me that a
Police Officer by default has the power to carry firearms, make warrantless arrests, using deadly force, etc etc etc
a
Peace Officer appears to have these powers, but the powers only apply by default when on duty, and the agency employing the officer has the power to determine if the officer can exercise off duty powers or even some on duty powers (basically an
ala carte cop where the agency gets to pick and choose which powers to afford a peace officer)
For instance look at this wikipedia page..
http://en.wikipedia.org/wiki/Law_enforcement_in_New_York_City#City_government_agencies
It appears in New York city there are about a dozen different law enforcement agencies belonging to several city departments, but that most of them like the Sanitation police and homeless service police are armed only with a baton (no firearm) but have full police powers such as the ability to write tickets and make arrests. also it appears new york's auxilary police officers are considered peace officers and that under an emergency declared by the mayor that the firefighters can become peace officers temporarily. what a strange set-up.