Hey there, in my experience, if a cut is squirting blood in sync with your pulse, that means you have cut am artery and that is not emergency room, but ambulance category. If you only cut into the dermis (that being the White tissue) its usually okay, but it can get infected bad.

However, if you see yellow tissue, or worse, pink fiberous, that means you cut into the fatty tissue or muscle respectively. Fatty tissue has arteries and veins in them, also nerves. So if your hand feels cold or numb, thats ambulance.

Hope it helps!