The video only shows him grabbing the nearest tool.
If you search around the room, you'll find other things like a wire cutter, cleaver, and scissors.
You can also find a metal rod, and use it on a stove to have a instrument you can use to close the wound after you cut off your finger, leading to another, very agonizing moment, but being much less painful for Ethan later on.
Keep in mind, this moment can be skipped entirely, the player can just waltz right out of the room. Infinity Ward, where as Deacon put it "Basically asks you 'are you going to pussy out on this?'" just issues a challenge for the hell of it, and the only purpose of it is to make you > HATE < the bad guys.
With Heavy Rain though, the plot device of it isn't to make you angry at the Origami Killer, but makes you actually feel > SYMPATHY < for what Ethan is going through to save his son.
Thinking about it, they ARE doing the same thing, but unlike feeling emotions like Sympathy or Empathy, Call of Duty tries to fuel you with Rage and Anger.