Equivalence of two definitions of substitution
The following remark from definition 2.1.15 gives you a clue:


  In the third clause it is not needed to say "provided that $y \not \equiv x$ and $y \not \in \mathsf{FV}(N)$". By the variable convention this is the case.


Since you can't make substitutions, it means that there is some violation of the variable convention (§2.1.13):


  If $M_1, ..., M_n$ occur in a certain mathematical context (e.g. definition, proof), then in these terms all bound variables are chosen to be different from the free variables.


In the meta-level expression

$$((λx.x)\ [ x := y ])\ z,$$

let us denote $M_1 = (λx.x)$ and $M_2 = x$. Notice that $x$ is bound in $M_1$, but free in $M_2$. That violates the variable convention, so you can't use definition 2.1.15, because it relies on that convention.