"imperial core" isn't a phrase we made up. It refers to World Systems Theory, a theory of international relations invented by a guy named Immanuel Wallerstein which argues that imperial "Core" countries (think the traditional "developed" or "first world" countries. Mainly the US and Europe) have a particular extractive, colonial relationship with "Periphery" countries (think poor, raw material exporting, rentier states like Kyrgyzstan or Nigeria).

Then there are semi-periphery countries which are still tied into the imperial core in some way, but have enough sway economically and geopolitically to kind of stand on their own. They have a different kind of relationship to the imperial core, compared to the periphery (these would be the BRICS countries, largely).

That's a gross over simplification, but hopefully that answers your question.

Edit: Here's a really good explanation of World Systems Theory that goes into more depth