The answer is to do what your spiritual father says.
In practice, I have seen many different instructions. In traditional Greek and Russian practice, as I have been taught, "no oil" means "no oil" of any kind. This means not to cook with oil, not to add oil to one's food, and not to buy food made with oil. It does not mean to avoid foods that naturally contain some oils. That being said, I certainly am familiar with the fact that some clergy in various jurisdictions in this country say that "no oil" means "no olive oil". To me, switching from olive oil to vegetable oil for the fasts is not much of a sacrifice.
Protopresbyter Alexander Lebedeff, the most senior cleric of ROCOR, makes some good comments on the subject here:http://www.orthodox.net/articles/fasting-and-oil.html
I also don't understand the allowance of beer or other forms of alcohol on "no wine" days, however. According to the instructions of our spiritual father, "no wine" = "no alcohol", but I understand that beer is permitted on "no wine" days in Slavic practice and perhaps in other local traditions. Beer seems to have been consumed throughout Lent in monasteries in the West, but this may have been by necessity on account of poor sanitation and poor nutrition (lack of drinkable water and lack of sufficient food).