Received an almost immediate message back from payfone:

"Hi [my name],

While we should explain more clearly that these services are used to protect consumers from fraud, with one's consent, you can set up what you are asking about with your phone company. We can send you the procedures to do that. I'd like to go a step further and see if we can just opt you out entirely across the board. I should also note that we do note store any current or historical personally identifiable data. Hold on for more info."

As an aside, I didn't provide [my name] with my request - of course they looked it up based on the phone number provided.