The burden of proof is on the person making extraordinary claims. Our friend the NYU psychologist has made an extraordinary claim and provided no evidence. Without evidence he expects his audience to accept hearsay as fact.
The problem with your standard is that it presupposes one (but not the other) as extraordinary. And once again, it assumes that this is even an either/or to begin with. I'm fairly skeptical about the idea of demonic possession, but can't definitively rule it out, either. But at the end of the day, something can work even if we don't understand (or are wrong about) why it works.
One claim - The patient is suffering from a psychiatric/psychological malady for which we have no easy explanation or efficacious treatment. The other claim - The patient is host to a demonic entity who is solely responsible for their suffering and whose presence can only be removed by an ordained member of the correct kind (Roman Catholic) of clergy acting according to the will of the correct (Roman Catholic) God. If someone gets exorcised while also receiving appropriate psychological care and they feel better - fine by me.