To which game-based security definition is Perfect Secrecy equivalent?
Perfect secrecy corresponds to semantic security. Or rather, semantic security is an adaption of perfect security for computationally limited adversaries, but a variant of the same definition also works for unlimited adversaries.

In other words, consider a game between an adversary and a simulator. The simulator chooses its key material. The adversary may submit messages and get their encryption in return. He then chooses a distribution over plaintext messages (all of equal length) and a binary predicate $f$, such that the probability that $f(m)=0$ is $1/2$. The simulator chooses a message according to the distribution, encrypts it and sends the challenge to the adversary. The adversary must now determine $f$. His advantage is the difference between his probability of guessing $f(m)$ correctly and $1/2$.

You need to be a bit careful with your encryption oracle and challenge ciphertext encryption, so that key material is only used to encrypt once. Once that it done, it is clear, as you say, that the encryption oracle doesn't help an adversary.

With a small amount of work, it now follows from perfect secrecy that any adversary in this game will have advantage exactly $1$. Also, if any unlimited adversary has advantage exactly $1$, it must follow that you have perfect security.

Note that if you try to adapt chosen ciphertext attacks, no reasonable definition will result in the one-time pad being secure, since it isn't secure against these attacks.