The hint is correct for the word, it just doesn't apply in this sentence. Yağı would also be the accusative of yağ -- but it is not that here because (1) a possessive-var sentence is about ownership, not location, and (2) you cannot use the accusative to mark anything here because there is no direct object when there is no verb.
Dear, if "yağı" stood alone then "-ı" clearly means: "the". However, here "-ı" indicates possession linked to "-ın" of "onların".
Possession mark of "onların" and "onun" which is: "-ı" can be confusing because it looks like the Definite mark. Other possessive marks are clear to be possessive such as "yağım/ yağımız ...".
Let's say you want to talk about a specific oil; ...ugh I forgot how to say that sorry..lol... Good luck
Değil and yok can both used as: "not" and "no".
But yok can have another special meaning; which is: "doest not exist" and "none".
in this course they are teaching us to use "yok" when we are speaking about the existence of something, while using "değil" for other negations.