Many often associate Proof of Work, Proof of Stake, etc with blockchain consensus algorithm/mechanisms.

I find that association to be inaccurate. Yes, proof of work / proof of stake are mechanisms by which leaders are elected to validate transactions. It is a crucial part in building consensus but not the only thing. For example, in bitcoin blockchain, the longest chain wins -- this to me is also considered as a crucial part in building consensus across the nodes.

As a blockchain nazi, to me, it only makes sense to call the distributed consensus algorithms such as Paxos, Raft, PBFT, etc as consensus algorithm/mechanisms because of its theoretical/mathematical underpinnings.

Why then is Proof of Work, Proof of Stake used interchangeably with Consensus Mechanisms/Algorithms?

From my understanding, as you add blocks there is a chance of the block chain temporarily splitting.

E.g. Block A has just been added, but at the same time block B was added. This creates a split in the chain, Chain A and Chain B. Blocks are continually added onto both, but eventually the network agrees on 1 chain and the other chain is discarded. Thats why you have to wait for confirmations when you deposit ether on to exchanges, they are waiting for the network consensus that the deposit has actually happened.

So it is a mechanism, although not the best one to form consensus. An example of blockchain that uses PBFT that you mentioned would be zilliqa.

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