When staking to the payment address is one effectively sending ADA to another wallet?
I think this is an interesting question because one asks if it could then be added to a wallet application like Daedalus, or something else..
A wallet consists of two sets of keys {private and public} and the addresses derived from them. There are different types of addresses like Payment address and staking address. Payment addresses can be used for receiving payment through transactions that require fees. The reward address of a wallet can not be used for receiving payment.
Overall we can say the public and private keys are the core components of a wallet.
Staking does not involve sending any ADA out of wallet.
If you want to understand how payment address, public pvt keys etc you can read this chapter https://docs.armada-alliance.com/ai-blockchain-edu/wallets