I think Ayoob was writing more about the times where he'd get his PD chief's documented authorization to assist another LE agency in a different jurisdiction. He'd be off duty but on "official business".
Clyde
PS; Years ago, I had a little "dust up" with a guy who claimed to be a US Treasury agent. He had a federal shape badge & creds. I was working security & found him in a back parking lot(private property) with a unknown female.
The "T man" was quite irate & became aggressive. I calmly explained I could contact the local sheriff's office & we'd settle the whole matter. The angry guy took off after I said that.
I recall him screaming; "
I can go where I want! I'm a federal agent!".