What is the benefit of zero knowledge identification (Fiat Shamir) over a standard digital signature?
Solution 1 has some weakness when the verifier is malicious. If the prover's private key $a$ is used for decryption, Solution 1 provides the verifier with a decryption oracle, i.e., a malicious verifier can decrypt any ciphertext encrypted with the public key $v$. On the other hand, Solution 2 reveals nothing about the private key $a$ because the random number $r$ (which masks the private key $a$ when the verifier sends a bit 1) is chosen by the prover rather than the verifier.