On definition of IP class
In the definition of class IP(k), the interaction always ends with prover’s message.  Otherwise there is no use for the last message (why?).

The situation is different for the definitions of public-coin interactive proof systems (also known as Arthur-Merlin proof systems), where sending a message from the verifier to the prover is the only way for the verifier to generate a random bit string.  In Babai [Bab85], AM(k) is defined by the verifier (or Arthur) sending the first message and MA(k) is defined by the prover (or Merlin) sending the first message.

[Bab85] László Babai.  Trading group theory for randomness.  In Proceedings of the 17th Annual ACM Symposium on Theory of Computing (STOC), pp. 421–429, May 1985.  DOI: 10.1145/22145.22192.