Partial Proof of Work in Pools
It depends on the payout system that they are using. Most payout systems use some sort of payment based on shares. A share is a Proof of Work that a miner submits to the pool. This Proof of Work is a "full PoW solution". But instead of having to meet the target for the Bitcoin network, shares just need to meet an easier target set by the mining pool.

Pools track how much work and how much hashrate someone has based upon the number, frequency, and difficulty of the shares submitted to them. They then use that to compute the payout amount. This computation depends on the payout system being used (e.g. PPS, PPLNS, etc.). The different payout systems calculate the final payout in different ways, but they all use this concept of shares.

The shares themselves are not categorized by their value. A pool knows what difficulty a particular miner is mining at so all shares are treated as being at that difficulty even if the value meets some other difficulty. The hashrate can be inferred from the number and rate of shares being submitted at a particular difficulty; the value of the hashes are not important so long as they meet the target.