In the table of quadratic residues modulo $13$, $1^2\equiv 1\pmod {13}$, while $12^2\equiv 1\pmod {13}$.
It is stated that the full list of QR modulo $13$ is $\{1,3,4,9,10,12\}$
The definition given below, and the examples above suggests that the below definition is unclear and may be wrong:
A nonzero number that is congruent to a square modulo p is called
a quadratic residue modulo p. A number that is not congruent to a
square modulo p is called a (quadratic) non-residue modulo p. We
abbreviate these long expressions by saying that a quadratic residue is
a QR and a quadratic non-residue is an NR. A number that is congruent
to $0$ modulo p is neither a residue nor a non-residue.
I feel that the correct definition should be:
A nonzero number that can occur as square modulo $p$ is called
a quadratic residue modulo $p$, else non-residue modulo $p$.
The reason is that the given definition seems to imply that the number (let, $x$) and any other integer's (in the set $\{0,\cdots,p-1\}$, let $y$) square (i.e., $y^2$) have the same residue modulo $p$. By that implication, it is not correct, even if take (apart from $1\equiv 1\pmod {13}$) that $3\equiv 9\equiv 3 \pmod {13}$, as the same does not apply to $4,10,12.$