On the definition of Error-Correcting Codes
Here is what the remark says:

A code $C \subseteq A^n$ is $t$-error-correcting if for all $x \in C$ and $y \in A^n$ such that $d(x,y) \leq t$, all $z \in C$ other than $x$ satisfy $d(x,y) &lt; d(z,y)$.

When is a code not $t$-error-correcting according to this remark?

A code $C \subseteq A^n$ is not $t$-error-correcting if there exist $x,z \in C$ and $y \in A^n$ such that $x \neq z$ and $d(z,y) \leq d(x,y) \leq t$.

Given $x,y,z$, either $d(z,y) \leq d(x,y)$ or $d(x,y) \leq d(z,y)$, and so, since the other constraints on $x,z$ are symmetric in $x,z$, the above definition is equivalent to

A code $C \subseteq A^n$ is not $t$-error-correcting if there exist $x,z \in C$ and $y \in A^n$ such that $x \neq z$ and $d(x,y),d(z,y) \leq t$.

This is exactly Definition 2.