The DM resolves the move as normal
Looking at the situation "I stab the fighter, who I'm standing right behind and who does not suspect me" definitely fulfills the trigger for backstab When you attack a surprised enemy with a melee weapon. So, that means the Thief succeeds in this first, sneaky attack and does damage to the fighter. If he does that though, remember, the 'bonuses' that come from rolling sneak attack don't come into play. So, having backstabbed his ally, the Thief is now in melee with the fighter and "I run to grab the ruby" no longer fits in the fiction.
If, on the other hand, the Thief chooses to roll+Dex to gain an extra effect from Backstab, the fighter is well within his rights to try not to get stabbed and Interfere. A success makes it more likely that the Thief fails. A failure gives the DM an opportunity to make a move which could, if they desire, derail the pvp. The same applies to hack and slash and with sufficient moves being made, it is almost inevitable that the DM will have his chance before the fight proceeds too very far.
As for whether the moves are 'meant' to be used against PC's, I would simply compare the trigger for Hack 'n Slash "When you attack an enemy in melee" with Parley "When you have leverage on a GM Character and manipulate them." Were the other moves supposed to be "NPC only," they would have been called out as such.
As a GM, you have a simple agenda:
Portray a fantastic world
Fill the characters’ lives with adventure
Play to find out what happens
Is the situation you described fulfilling that agenda?
Portray a fantastic world: Absolutely! Piles of treasure, presumably at the end of some harrowing dungeon set the players and their characters solidly in a fantasy world.
Fill the character's lives with adventure: Absolutely! Though they've taken some of the initiative for themselves, a fight over the treasure is definitely more interesting and adventuresome than simply counting coins. That said, this is the agenda that best supports interceding with some kind of outside complication, should you not want the fight to go too far.
Play to find out what happens: Absolutely! The betrayal probably took you as much by surprise as it did the poor fighter. Enjoy as drama completely unplanned plays out in front of you.