When knowledge soundness implies soundness
The difference is that in Section 4.5, knowledge soundness (i.e., extraction) is required to hold only for every $x\in L_R$, and so there is no requirement at all for the case that $x$ is not in the language. In contrast, in Definition 9 of Lin03, the knowledge soundness (i.e., extraction) is required to hold for all $x$. This implies soundness since you cannot extract a witness for $x\notin L_R$, and so the verifier cannot accept in this case (except with probability of the knowledge error).
Note that this is orthogonal to the question of whether or not $P^*$ is efficient (i.e., whether it's a proof versus an argument).