I understand how Proof of Work (PoW) works now, and am beginning to look at different Proof of ... systems like these:

From what I have read so far, it sounds like the term "Bitcoin Mining" means Proof of Work in the Bitcoin system. But then I'm not sure if mining is an inherent part of all these other systems and if it works in a similar way. If not, I'm wondering what the general meaning of mining is across these different "Proof of ..." systems.

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The term mining is most specifically applied to Proof of Work consensus mechanisms. This is because verifying a bitcoin block is a rigorous process of hashing the block header until you find the block header hash that matches the current difficulty goal. New bitcoins are generated in the process (incentivizing mining) and hence the term mining.

Proof of Stake - In this consensus protocol, all the hard limit on the coins are outstanding on day 0 of the blockchain. This can achieved mostly through ICO. Now, the validators in this protocol stake their coins as a collateral in the network. The collateral serves as a guarantee that the validators do not act in a malicious way as they can lose their collateral. Every validator gets selected at random to verify a particular block. The chance of getting selected is proportional to the amount of money that is staked. The validators collect only the fee of the transactions in the block. The validators in Proof of Stake are called forgers or minters.

Proof of Ownership - This is not a consensus protocol. It is a byproduct of the blockchain that allows you to track the owners of certain information. The most glaring example would be colored coins. This is stored in the transaction as a metadata using a OP_RETURN opcode.

Proof of Existence - Again, this is not a consensus protocol and is basically a byproduct of the blockchain that allows you to verify that a particular record or transaction exists on the blockchain. This can be verified by different parties independently.

Proof of Space - This is a consensus protocol that leverages disk space rather than computational power. To mine blocks using PoSpace miners invest disk space to the network, with the probability of mining the block proportional to the space allocated as compared to the total capacity of the network. A PoSpace is a piece of data that the prover send to the verifier to show that he has reserved a certain amount of space. Without reserving that much amount of space, there is no way the prover can send that data. One way in which PoSpace can be implemented is by using hard-to-pebble graphs. The verifier asks the prover to build a labeling of a hard-to-pebble graph. The prover commits to the labeling. The verifier then asks the prover to open several random locations in the commitment.

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