Don't worry. Once you start, you will see that core exam questions are similar to the ones that you will be dealing with during the first year. And a first year student in Harvard may fail the core exams of any other school even if he/she has passed Harvard cores. Every school has a different structure and first year sequences can really differ from school to school. Professors generally like to teach in their own way of thinking and ask questions in that way. For instance; Lar Peter Hansen's Econometrics course at Chicago is unique in US. It is an excellent course, but one can even think that it is not econometrics.
So don't worry about the cores at this stage.
By the way, they all have the same meaning, but i'd prefer to call it "core" for obvious reasons.