His adversaries were under the impression that the machine calling a particular phone after a reset would establish an Admin.

That wasn't quite true.

Under normal circumstances, the machine was quite able to find Harold and contact him at the nearest available payphone (contacting his cell directly might be too dangerous if it was traced). But Harold's assumption had been that if the Machine was compromised to the point where it needed to reset and reboot its system, its usual resources might not be available. So it would need a simpler method of contacting Harold to let him know of the problem. A method that still did not involve direct contact. If the Machine was in some way compromised, it was entirely likely that the government might have had a hand in it.

So the public library payphone was established as an emergency contact.

The others thought that whoever answered the phone would become the ultimate Admin. The truth was that the call was expected to be answered by the ultimate Admin (Harold). As Harold was limited in his ability to contact the Machine, and the Machine was careful in its contacts with Harold, there needed to be some safe way for the Machine to notify its Admin that it was in trouble. In case of such an emergency the Machine would call the designated phone. Harold would continuously monitor the line to be alerted if a call for help came in. If he could not intercept the call himself he could establish contact later to assist the Machine in restoring its systems to optimal.

The Machine didn't really care who answered the phone, only that its Admin knew it needed his help.

And that call for help was the one Harold planned on answering.