an empire is a state where the core ruling population exercises control and extracts resources from a periphery which is composed of people other than the core group (linguistically/culturally/ethnically/religiously distinct)

I don’t think this is quite general enough—he later brings up the example of the Athenian Empire, which wasn’t based on linguistic, ethnic, etc. distinctions. And the Roman Empire changed every aspect of its “core group” multiple times over its existence.

I'd define an empire instead as a state where a stratum of alien institutions has been imposed on top of a substratum of local institutions with which it didn’t coevolve and therefore has no organic relationship.