Yes. We aren't talking past each other anymore. If you aren't composing a "salted hash" out of a cryptographic hash function and a salt, but are instead using bcrypt, scrypt, PBKDF2, or Argon2, you have nothing to worry about. Just to complete your understanding: the salt --- randomizing the hash --- has very little to do with what makes bcrypt a good password hash.
You'll avoid this confusion in the future if you don't refer to bcrypt as a "salted hash". Salted hashes are the technology bcrypt was invented to replace.
OP here: I am using bcrypt to hash passwords, but I struggle to understand how not salting i.e. not randomizing it against a stored secret would still be safe for storing short strings. Be that as it may, I'm glad to hear I'm not doing anything horribly wrong when building login systems.
You'll avoid this confusion in the future if you don't refer to bcrypt as a "salted hash". Salted hashes are the technology bcrypt was invented to replace.