Review the description of Lamport Causal Clock in the article. Note that it carries “additional info” (additional to the example diagram). This “additional info” is what establishes the structure needed for total order.
> whichever concurrent updates you don't choose are lost, and that violates consistency
They’re not lost! The concurrent updates not chosen are still part of the “list of operations”, but the update to the data/value itself may not be observable if the subsequent update (the update that was sorted to be first) updates the same data/value (eg. both operations update the same key in a key-value structure). If the two operations update different data/value, then both updated values are observable. This isn’t cheating, rather it works exactly as expected: it is eventually consistent.