I believe the books say the PTWP would be “born under a dying star.” The books also make a much closer case for John’s death/rebirth at the Wall being the moment TPTWP is born. Several of John’s men are crying as they stab him (salt), the steam escaping from his wounds makes them “smoke” in the cold northern air, and Ser Patrek has been smashed against the battlements of the castle by Wun Weg Wun Dar Wun, dying as his blood covers his sigil, which is a star*. Not sure if the TV show has all these elements (I’m almost certain they don’t have Ser Patrek, making the “dying star” part difficult), but they are definitely in the source material.
*a star that, as written, is a dead ringer for the Dallas Cowboys logo — of course Martin found a way to kill a Cowboys fan in his fantasy book.