Yes you understood it correctly, the function password_hash() will generate a salt on its own, and includes it in the resulting hash-value. Storing the salt in the database is absolutely correct, it does its job even if known.
// Hash a new password for storing in the database.
// The function automatically generates a cryptographically safe salt.
$hashToStoreInDb = password_hash($_POST['password'], PASSWORD_DEFAULT);
// Check if the hash of the entered login password, matches the stored hash.
// The salt and the cost factor will be extracted from $existingHashFromDb.
$isPasswordCorrect = password_verify($_POST['password'], $existingHashFromDb);
The second salt you mentioned (the one stored in a file), is actually a pepper or a server side key. If you add it before hashing (like the salt), then you add a pepper. There is a better way though, you could first calculate the hash, and afterwards encrypt (two-way) the hash with a server-side key. This gives you the possibility to change the key when necessary.
In contrast to the salt, this key should be kept secret. People often mix it up and try to hide the salt, but it is better to let the salt do its job and add the secret with a key.