They're not really that ambiguous.
No such thing. But 6 is RAW.
Mage Hand, PHB p256.
"You can use your action to control the hand. You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial. You can move the hand up to 30 feet each time you use it.
The hand can't attack, activate magic items, or carry more than 10 pounds."
Now read the section on oil, PHB p152.
"As an action, you can splash the oil in this flask onto a creature within 5 feet of you or throw it up to 20 feet, shattering it on impact. Make a ranged attack against a target creature or object, treating the oil as an improvised weapon."
Okay, so what roll are you going to make to pour oil onto a target? An attack roll. What can't you do with mage hand? Make an attack.
"But it clearly states I can 'pour the contents out of a vial'."
Yep. But it also clearly says you can 'manipulate an object'. A sword is an object. So why can't you use mage hand to attack? Because you're specifically not allowed to attack with the mage hand. Specific rules beat general rules. You're generally permitted to manipulate an object, except you specifically cannot use the mage hand to attack. You're generally permitted to pour the contents out of a vial, except you specifically cannot use the mage hand to attack.