Confusion about definition of homomorphic encryption
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.