There lies your problem, the spider is not "being revived" because it techincally does not die UNLESS it has a HP burn on it. It is the ONLY way to kill the frost spider. If at the moment of death, it does not have the HP burn debuff, it will regen to 50%. Yes the animation is a little confusing because it acts like it is being killed, but not the case.

Mark of Fear [ATK] (Cooldown: 4 turns)
Attacks all enemies. Has a 50% chance of placing a [Freeze] debuff for 1 turn. Enemies killed by this skill cannot be revived. 

The ability should work, since the same mechanic works when killing it with Zavia's (A3) abilty for me. Smells like a bug to me! 

im sorry but that´s not correct. i dont have hp burn toons, but i can kill it with Arminger.