State Police means they have jurisdiction in the entire state.
State Patrol may or may not have limits on jurisdiction.
Arizona Highway Patrol has jurisdiction in the entire State, but they let locals handle the non traffic stuff, because they couldn't figure out if there was a domestic if the bloodied participants were standing in front of them.
I don't know if it has changed, but back in the 90's when I was a Deputy in Ohio, the Ohio Highway Patrol had no jurisdiction outside the fences of the Highway (the highway includes berms, sidewalks, and of course, the travel way, that vehicles travel on).
Many of our guys were surprised to discover that back then (again, I don't know if it has changed), OHP or OSP, depending on who you ask, did not even have the authority to carry off duty.
One of their guys took a plea in a domestic case to low class misdemeanor unlawful carrying a firearm, which saved his job.
Jay