Well your Wikipedia link was about cryptographic hash functions so I figured that's what you intended to refer to.
So when you wrote "key-based hashes", I interpreted that as meaning a cryptographic hash-like function with key input, e.g. HMAC the "Keyed-Hash Message Authentication Code".
Modulo 13 is different than rot13. Modulo 13 is actually a hash function, whereas rot13 is a permutation.
If rot13 took a key (e.g. if it were rotN instead) it would make a primitive cipher. But it doesn't, so it behaves like a cipher that is always used with a fixed key or a cipher the key is already decided in the context of discussion.
The process of applying a specific key to a cipher is called "keying". So just to make things even more confusing, we could perhaps refer to rotN<N = 13> (AKA "rot13") then as a "keyed cipher".
So when you wrote "key-based hashes", I interpreted that as meaning a cryptographic hash-like function with key input, e.g. HMAC the "Keyed-Hash Message Authentication Code".
Modulo 13 is different than rot13. Modulo 13 is actually a hash function, whereas rot13 is a permutation.
If rot13 took a key (e.g. if it were rotN instead) it would make a primitive cipher. But it doesn't, so it behaves like a cipher that is always used with a fixed key or a cipher the key is already decided in the context of discussion.
The process of applying a specific key to a cipher is called "keying". So just to make things even more confusing, we could perhaps refer to rotN<N = 13> (AKA "rot13") then as a "keyed cipher".
:-)