I recently saw an offer to "Buy Bitcoin Core with your credit card". Is this "Bitcoin Core" the same thing as BTC?
Kind of. People who call Bitcoin "Bitcoin Core" are referring to Bitcoin. They will also use the name "Bitcoin Legacy". but calling Bitcoin "Bitcoin Core" or "Bitcoin Legacy" is misleading.
Bitcoin Core is the reference client for Bitcoin. But it is wrong and misleading to call a coin by the name of its reference client. For example, we don't call Bitcoin Cash "Bitcoin ABC" even though its reference client is Bitcoin ABC. Nor do we call Ethererum "geth" even though its reference client is Geth.