Correct me if I m wrong in understanding Ethereum

Ethereum has two type of accounts - Externally owned and Contract accounts.

Anybody can create a contract and put it into the blockchain, and it will have its own address.

Now if I as a node on the same blockchain, want to make use of that contract, I will supply the parameters and gas price of it.

Heres the part I am not sure about - Now just like Mining, the miners will fight for POW/POS puzzle, and whoever wins it, is going to fire the smart contract code and earn the gas reward. Am I right here?

Also what is going to be saved in the blockchain database now? As in whats the proof that the node actually processed the smart contract?