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.

While sometimes "Bitcoin Core" is used to refer to Bitcoin, as mentioned in other answers, there is actually also an altcoin that is called "Bitcoin Core (BTCC)". It's currently trading under $1/BTCC, though.

Yes Bitcoin Core is BTC. This is the original Bitcoin.

The other forks that were created recently are Bitcoin Cash (BCH) and Bitcoin Gold (BTG), among others. Their symbols are different.

Bitcoin, Bitcoin Core and Bitcoin Legacy are all identifiers for the first Bitcoin (BTC/XBT). Other forks such as Bitcoin Cash, Bitcoin Gold and many other claim to be either the real Bitcoin or a better one. This results in Bitcoin being often referred as Bitcoin Core or Bitcoin Legacy.

Yes, Bitcoin is the same as Bitcoin Core and Bitcoin Legacy. This naming came up because of Bitcoin forks such as Bitcoin Cash and Bitcoin Gold.

Also BTC = XBT.