The definition of ZKP is (1), but that implies (2), as explained in this paragraph from the Wikipedia article:
>For zero-knowledge proofs of knowledge, the protocol must necessarily require interactive input from the verifier, usually in the form of a challenge or challenges such that the responses from the prover will convince the verifier if and only if the statement is true (i.e., if the prover does have the claimed knowledge). This is clearly the case, since otherwise the verifier could record the execution of the protocol and replay it to someone else: if this were accepted by the new party as proof that the replaying party knows the secret information, then the new party's acceptance is either justified—the replayer does know the secret information—which means that the protocol leaks knowledge and is not zero-knowledge, or it is spurious—i.e. leads to a party accepting someone's proof of knowledge who does not actually possess it.
That is, if the proof were convincing to the entire world, then non-possessors of the knowledge could just replay they proof without having the knowledge.
>For zero-knowledge proofs of knowledge, the protocol must necessarily require interactive input from the verifier, usually in the form of a challenge or challenges such that the responses from the prover will convince the verifier if and only if the statement is true (i.e., if the prover does have the claimed knowledge). This is clearly the case, since otherwise the verifier could record the execution of the protocol and replay it to someone else: if this were accepted by the new party as proof that the replaying party knows the secret information, then the new party's acceptance is either justified—the replayer does know the secret information—which means that the protocol leaks knowledge and is not zero-knowledge, or it is spurious—i.e. leads to a party accepting someone's proof of knowledge who does not actually possess it.
That is, if the proof were convincing to the entire world, then non-possessors of the knowledge could just replay they proof without having the knowledge.