It's established that skin-spies do not understand paradoxes and humor. When Achamian joins the Skin Eaters, it's quickly established that Soma often misses the jokes that the other veterans tell. Soma is later revealed to be a skin-spy, so he was apparently a skin-spy right from the start.
Skeaös is eventually revealed to be a skin-spy, but it's never known when the switch occurred. However, when Conphas talks with Skaeos while climbing the vast staircase to the Andiamine Heights, Conphas is surprised that the elderly Skaeos exhibits no physical discomfort at all from the grueling climb, showing that the switch has happened before this time.
Benjuka is said to challenge one's spirit rather than intellect, which is why the intelligent but insecure Achamian usually loses to the more confident Xinemus. In one scene, Achamian makes a firm decision on an unrelated matter just before making a move in the game. The surprised Xinemus laments that all of his recent gains in the game have been undone by the move. Achamian's sudden burst of spiritual strength has made him (at least temporarily) a better benjuka player.
In Koringhus's POV chapters, he thinks of himself solely as "the Survivor," and his son also thinks of him with this same appellation. As Dunyain, they can practically read each other's thoughts, so it's perfectly natural that the son would pick up on Koringhus' self-applied name.