You're on actually correct.
Here's how it works... That seee phrase plus an optional extra word you can add goes through a robust process (which I think is called PBKDF2) that involves a specific "salt" per say (the word "mnemonic" is used here). This process spits out a long 512-bit string.
This 512-bit string is then split in two. The first half is your actual Master Private Key—the ultimate secret. The second half is the Master Chain Code. This code is vital for creating all your future addresses in a secure organized way ensuring your wallet functions smoothly. It's not part of the private key itself but a kinda like a companion.