I'm not aware of a legal definition of "police" that prevents private corporations from claiming to have a "police force." Seems to me it's marketing and security theater- use the word to obtain "respect" (fear?) from the public; demonstrate (theatrically) that the company "cares" about safety. (That said, Amtrak isn't exactly a private corporation...)
Basically, regardless whether it's an Actual Police force, or a pretend one, you need to be aware of your rights and never acquiesce simply because you "have nothing to hide."
In the US sworn law enforcement officers have legal powers that private security does not, such as the authority to make arrests. Private security can't call itself police, impersonating a police officer is a crime.
"Police impersonation is an act of falsely portraying oneself as a member of the police, for the purpose of deception."
"...for the purpose of deception" being the operative phrase. Campus "police" aren't attempting to deceive anyone, they're attempting to enforce university rules.
IANAL, but deception here means, to convince someone that you're a sworn law enforcement officer, which is what a private security service would be trying to do. The exception is there so that for example an actor dressed as a police officer, or someone in a halloween costume, is not a criminal because they are not trying to convince a member of the public that they truly are a police officer.
Basically, regardless whether it's an Actual Police force, or a pretend one, you need to be aware of your rights and never acquiesce simply because you "have nothing to hide."