I believe it. People born after the payphone era might not realize that the telephone cord in this case wouldn't be the cushy plastic home-telephone cord such as you can find by googling "strangled with telephone cord," but rather a "payphone armored cord", which looks almost like a "metal-clad electrical cable" except more flexible and stretchy (such that you conceivably could get a nasty pinch from it when it contracted again).

Also, while I believe a payphone-cord pinch could draw blood, I don't believe any hospital staffer could mistake an actual stabbing wound for such a pinch. So the guy couldn't have actually gotten stabbed. Although I guess he could have been covering up for the drunk friend who dared him to pinch his neck in a payphone cord...