Do shares' difficulty really show luck?

The amount of shares shows the effort put in, while the difficulty of those shares indicates the luck involved. Is that accurate?

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.

I completely agree with the explanation about shares, but I don't think you can compare difficulty and luck. Difficulty is directly proportional to hash rate or the amount of hardware in mining. I have seen in GPU mining a pool selection for a certain difficulty, so that weak and powerful graphics cards can select different pools. According to mathematical formulas the luck of any mining equipment tends to 100%, but time plays an important role here. Powerful equipment can mine 1 block solo in a year for example, while weak equipment can mine 1 block in 100 years. But 100 years the equipment can not mine.

Mining difficulty and luck are not even close. Mining difficulty have rendered many miners useless, because they just can't keep up anymore, if there is a way to lower difficulty then those outdated miners will be capable again. Let's say that a reset is possible on Bitcoin mining difficulty, you will see that CPUs can mine BTC again, someone somewhere will update the mining software that are compatible with CPUs to mine BTC. Once a miner is outdated it is over, there is no coming back, this is why I laugh at people spending too much money on small Asic miners to solo mine BTC, yes blocks are solved luckily, but that's like 1 out of 1000 Asic miners solved a block luckily. This is not something you can bet on.