You explained it just right.
NHL players can be in the game after their first NHL appearance.
AHL players can be in the game after they sign their PHPA waiver form. League reps go through team-by-team asking the players to sign the form. It usually takes them through Dec. or later to get to every team. The players aren't required to sign them either, and then they just don't end up in game. Joel Armia hasn't the past 2 years from what I hear.