The number of shares proves the work, and the difficulty of shares proves the luck. Is that true?
Your question is not clear, however, it's both share count and share difficulty that prove the work. I don't know what prove "luck" means to you, so I can't answer this part.
Now, back to "work": how would a mining pool know that your miner is submitting work and is entitled to receiving rewards? It's simple, every share you submit that meets the pool difficulty is recorded and you get rewarded accordingly, depending on the pools method, you may get paid for every share regardless, others have different payment methods which differences "slightly" like PPLNS, but eventually, all pools pay you for the shares that you submit above x difficulty, it's just that some would pay you wether they hit a block or not -- while others will only pay you if they hit a block and your shares were present in the N period.