Confusion about definition of homomorphic encryption I am trying to better understand homomorphic encryption, but I feel like I keep getting inconsistent information in the papers that I am reading. 

One of the papers I am reading says the following:


  Earlier, homomoprhic encryption was defined as a form of encryption
  where a specific algebraic operation performed on plaintext is
  equivalent to another algebraic operation performed on its ciphertext


Meanwhile, another paper says the following:


  For now, a homomorphic cryptosystem can be thought of as an oracle, or
  black box, that, when given two ciphertexts and an operation, returns
  an encryption of the result of that operation on the two
  corresponding plaintexts.


The wording of the first one makes it seem as though the operation does not have to be the same (it says "another"), but the wording of the second one makes it seem as though the operation IS the same.

Can somebody clear up which one is correct?
The operation does not have to be the same. For example, with Paillier, we multiply ciphertexts to get the addition of the plaintexts.

That said, I think what the 2nd quotation is saying is that the operation that is passed to the oracle is the desired operation in the plaintext domain. The oracle knows how to translate that operation into something that, when performed on the ciphertexts, results in the desired operation in the plaintext domain.