Petroleum means crude oil, on both sides of the Atlantic. Petrol, a product manufactured from petroleum, is fuel for cars. It is called gasoline in North America.
Oil is a much broader word. Sometimes it is a synonym for petroleum. There are vegetable oils, such as olive oil, used in cooking or as ingredients in foods, or made into soap and other products. Mineral oils are various substances made from petroleum. I see that your Cambridge Advanced Learner's Dictionary gives diesel oil as an example in BrE. That would be called diesel fuel in America. And of course, various lubricating oils are made from petroleum.