To my understanding BTC and bitcoin core are the same, when mentioned against S2X/B2X. Is that the case? (Bitcoin Core is BTC in the post segwit2x world?)
Bitcoin Core is a specific implementation of the Bitcoin protocol, specifically the one in the Bitcoin GitHub Repository. It is the evolved version of the original client Satoshi wrote, in C++. It is also the reference implementation of Bitcoin.
But Bitcoin Core is not the only Bitcoin node software, there are others written in other programming languages and with other extra features, for example there is bcoin, btcd written in Go, Bitcoin Knots, Bitcore, etc.
When being compared to S2X, Bitcoin Core is sometimes used to refer to the Bitcoin protocol defined by the Bitcoin Core reference implementation - i.e. Bitcoin itself, to avoid confusion. Bitcoin Core is also sometimes used to refer to the developers of Bitcoin Core, but it must be noted that Bitcoin Core is not an organisation, it is a piece of software.