Call it Master private key not Master key..
Master key is correct, it means master private key.
It’s not only the master private key that generates the private keys, it needs the chain code to form “ extended private keys” , it isn’t like chain code is created separately, they are always together but when you split them, you get a prv - chain code.. together they form xprv(extended private key)
Do not mind this definition, extended private key can be referred to as master private key. Some people can refer to it as parent private key.
So anywhere you see people referring to master private key, know that they can use extended private key to refer to it. The master key, chain code and index number generate the child private key which is the private key tyr OP is referring to.
Hence, the xprv can generate all private keys (in the sense that a parent key can generate a new private key, which can then act as a parent key again) and also generate their corresponding public keys, while the xpub can only generate public keys of non-hardened child keys.
xprv is for legacy addresses
yprv is for nested segwit
zprv for native segwit
The x and z are X and Z for multisig wallets.