Byzantine Fault Tolerant consensus among known validators
A third real, widely used approach runs consensus among a known, fixed set of validators who explicitly exchange votes with each other and finalize a block once a qualifying supermajority, commonly two-thirds, agrees. This is often called BFT-style, or Byzantine Fault Tolerant, consensus, and it's used by a number of real blockchains and enterprise systems, sometimes on its own and sometimes layered together with staking.
Because the validator set is known in advance and votes are exchanged directly rather than inferred from computational races, this style of consensus can reach finality, meaning a block is confirmed and irreversible, very quickly and predictably. The tradeoff is that it generally requires knowing who the validators are ahead of time, and it tends to work best with a validator set that isn't extremely large, since every validator needs to communicate with every other one to reach a vote.