What if somebody else (i.e. Not Carlisle) turned Edward? Maybe a nomad the year before?
Well.
I imagine Edward would get on the eating criminals train sooner and never get off. He's turned into this demon by what he'd view as a monster and told he must eat people to survive. Edward is horrified at eating innocent people and likely comes to the conclusion that, if he must eat people to survive and cannot destroy himself or starve himself, then he'll eat wrongdoers.
As in canon, he tells himself that this isn't just a benefit to him, but it's making the world a better, safer, place in that he can take out things that go bump in the night. He'll be the monster who feeds only on monsters.
He abandons his creator as soon as he can and starts going about this as he did in canon. However, I imagine as in canon, he soon becomes depressed and slowly realizes that no matter the excuses he's giving himself he's ultimately doing this for his own pleasure and is certainly no better than they are.
However, there's no hope for Edward as there's no Carlisle and Esme to return to. Edward likely exists in a depressed fugue state and either tries to put himself into a coma by sinking to the bottom of the ocean, picks a fight with other vampires to get them to kill him, or vacillates between feeling great about killing wrongdoers, feeling terrible, and falling off the wagon and eating those even he admits aren't doing 'wrong'.
Edward spends his existence despising himself.
