In GotM Hairlock is killed by a wave of sorcery, and when it hits Tattersail she describes it as cold.
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"She saw a wave sweep past Hairlock's defenses, cutting him in half. His howl was more rage than pain, instantly muted as virulent power washed over Tattersail and she found her own defenses assailed by the sorcery's cold, screaming will as it sought to destroy her."
Presumably Tayschrenn's magic would not be cold, given that it is the Warren of Fire. Therefore, it could be assumed that that was a deflection of Rake's power by Tayschrenn. Calot on the other hand was killed by fire sorcery...
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"A blast of bright fire engulfed him (Calot)."
...which makes little sense, as Tayschrenn would have no cause to kill Calot. Another stone in the proverbial shoe.
As for Nightchill I had assumed that it was more how it was described in MoI, and the other version is just a GotMism. Nightchill wanted Dragnipur, and knowing that A'Karonys supected her motives, she killed him-ignoring the sequence of events as described in GotM-Tayschrenn, having been warned by A'Karonys, knew that it was Nightchill. And he therefore unleashed a demon upon her, I do not think he summoned it then, not for the first time, there was probably already a link between him and the demon, therefore summoning it did not require any significant amount of power that would otherwise weaken his defenses or assault .