English English speaker here: quite is often used both ways. While it is defined as meaning "fairly" it is usually used to mean a little more than that ("the weather is quite nice" means somewhere between "the weather is fair" and "the weather is nice"), it is also used to mean entirely but usually in a sarcastic manner where the juxtaposition of the entirelyness of your intent and the fairlyness of the actually meaning of the word stands out ("oh, you are quite wrong there"). Also, in combination with a negative it usually means entirely (as in "the job isn't quite done yet").
These are far from hard & fast rules though. In spoken situations the intended meaning is generally obvious from tone, in written text it can be quite ambiguous.
There is at least one use where the word means both at once: when it is used on its own: Person one: "The man is an utter 'king imbecile", Person two: "Quite" or P1: "WOOOOOOHOOOOOOO!!!!!!", P2: "Quite". Here it means "I agree (on a binary yes/no level) but I wouldn't put it that strongly". Similar to how "indeed" is often used.
These are far from hard & fast rules though. In spoken situations the intended meaning is generally obvious from tone, in written text it can be quite ambiguous.
There is at least one use where the word means both at once: when it is used on its own: Person one: "The man is an utter 'king imbecile", Person two: "Quite" or P1: "WOOOOOOHOOOOOOO!!!!!!", P2: "Quite". Here it means "I agree (on a binary yes/no level) but I wouldn't put it that strongly". Similar to how "indeed" is often used.