Why mining pools miners need to choose their difficulty? I read this and this.

It said: "The traditional method involves assigning members a work unit comprised of a particular range of nonce, the number that blockchain miners are computing for. Once the pool member completes the work on the assigned range, he places a request for a new work unit to be assigned."

From what I understand, the nonce range is the same for all miners and in order to proof their work, miner need to send the share with the difficulty fixed.

So I don't understand the interest to increase the difficulty for a miner. If he keep the minimum difficulty, why would he reduce his lucks to proof his work with a higher difficulty?
Having a different difficluty and thus have lower luck does not reduce the payout at all. Payouts are proportional to a miner's hashrate. So with higher difficulty shares, the payout will be higher per share which compensates for having fewer shares.

However a miner will want to use an appropriate difficulty level even though the payouts are the same. Although the payouts will be the same in theory, in practice, there is a difference. If a miner mines at too low of a difficulty, the mining hardware may spend a lot of time having to send and receive data from the pool server that it remains idle for so long it can't submit as many shares as it should so the payout isn't as high.

Conversely, if the difficulty is too high, the hardware won't ever be able to find a share so the payout will be basically 0. So a miner should choose a difficulty which matches his hardware so that it is at 100% utilization almost all the time but still be able to submit shares frequently.

The pool difficulty itself has no effect on the luck a miner has on finding a share at the actual network frequency. Pool difficulty is just a way for pools to keep track of how much work each miner is doing.