Burning Aura, specifically happens when a model is hist by a strike but before damage. However the attacker would still make a damage flip, The difference between Burning Aura and Burning Blood is that for Burning Blood to take effect, the defender actually has to be Damaged, so if the attacker flipped a black joker for damage they wouldn't suffer. However on Burning Aura, even if the attacker flipped a Black Joker on the Damage Flip, they would still suffer the effects.
Even though the attacker died on the Strike, they would still get the damage flip because the damage flip is an effect of the Strike.
Its also worded that way because Iggy has Martyr, if it only worked on damage, no one would suffer if he martyred the damage onto another Woe. Whereas with it being worded on being hit, he can slough off the damage onto another woe, while at the same time making his attacker suffer.