Bitcoin Proof of work puzzle inconsistency At this time, I am really confused about the Proof of work puzzle because as I have seen in some documents, there are 2 different ways to talk about them.

The first one I have seen is from some videos, and also the original of Bitcoin by Nakamoto: the answers to the puzzle will make the result of hashing the block header contains some number of zeros from the start consecutively.

Meanwhile, there are also some documents saying that the result of hashing the block header has to be below a difficulty.

So I wonder which one I can trust now. Could anybody tell me? Thanks in advance :-)
The Proof of Work is not really a puzzle that is solved.

For a block to have a valid proof of work, the hash of the block header must be less than a given target. This will result in the hash having some number of leading 0's. People will then simplify that to be "some number of leading 0's" when explaining PoW to people. But it is actually less than a certain target value.

The target value is used to determine the difficulty (it's an inverse relationship: as the target decreases, the difficulty increases). If you were to calculate the difficulty of a block in the same way with its hash, then the difficulty of a block must be greater than or equal to the network difficulty.