I have a theory on the scar around his neck.. I mean, he died of BEHEADING and the achievement we get is the story with Theseus who did it. So I'm guessing that what we did kind of was a parallel for his "fight" with Theseus, only, he broke the recursion spiral by fighting back and so the scar that exemplifies his passive "cowardice" in allowing the beheading vanished. (And maybe paved the way to annulling some contractual stuff later.. )
Or it has something to do with the other story, where he "cuts the thread" for that immortal guy whose name I can't remember.. Tithonus? and apparently takes on his shackles instead. Maybe those shackles are the "band" on his throat like a collar. It's a bit weird that Asterion himself seems very unconcerned with what happened and it really feels like a biased narrative viewpoint again.