I think you miss the point of security through obscurity. It's not about keeping the information itself obscure (in your example login information), but rather the method. For example, your password hashing mechanism. If you have a strong password hash function, you don't need to obscure which hash function you use (otherwise, open source software couldn't even exist in certain areas). However, if your security relies on you obfuscating your broken, home-made hash function that only hashes the first three letters of the password, you're not really secure. Security through obscurity is an attempt at securing an otherwise unsecure system by hiding or disguising the implementation.
That being said, obscuring parts of an otherwise secure system is fine as an additional layer, especially if you just want to deter script kiddies that always hammer the same endpoints