How is the staking module implemented in Ethereum? Is it part of the client/node logic? Any pointers to the code?

There is a deposit contract deployed on-chain, which feeds staked ETH into a “consensus layer” that deals with the staking logic. Good resources:

Robust Incentives Group @ Ethereum Foundation

https://rig.ethereum.org

aaa gotcha, that's helpful! so essentially the validator will stake their ETH in the deposit contract, and can prove that they have done so in the consensus logic which allows them to participate (am I thinking correctly?)

There is a deposit contract deployed on-chain, which feeds staked ETH into a “consensus layer” that deals with the staking logic. Good resources: https://github.com/ethereum/consensus-specs https://ethereum.org/roadmap/beacon-chain