what is a rigorous definition of "proof-of-possession"?
The closest notion used in the literature seems to be “proof / argument of knowledge” (PoK / AoK). See [GMR, BG] as well as Wikipedia for its definition. Indeed, giving out the key would be a proof of knowledge (but it's not of much interest to cryptography).
The part about “not sending the value to the other party” seems to be related to zero-knowledge (ZK). ZK requires the interaction yields no extra effect beyond the prover convincing the verifier. For example, if a ZKPoK is used to prove possession of a secret key with respect to a public key, then the verifier does not learn any extra practical information about the secret key after being convinced. See [GMR] and Wikipedia.