Remember salts don't need to be secret to do their job. The goal is to change the algorithm slightly (by adding additional input) for each user. That means you can't mass-precompute (rainbow tables), and just look up what matches, you have to break each user individually.

Your reasoning about how salts work is correct.

There's also something called a pepper which is another additional bit of input data, that is only stored in the app code (fixed for entire app). So an attacker who only manages to get a database dump would need to guess yet another chunk of data (making it near impossible). So a well-seasoned hash would be SLOW_HASH(pepper+salt+password).

Security is all about layers. Each layer protects a bit more, or prevents things from being easy for the attacker.

Edit: Don't do this yourself. Know it for the theory part - but then just use a well-vetted library to do it.