The small dihedral groups $D_1$ and $D_2$.

The small dihedral groups $D_1$ and $D_2$.

In M. Artin's book Algebra he wrote:
But I think this visualisation $D_1$ and $D_2$ is inconsistent and
confusing, because I guess $D_n$ could be associated with a subgroup of
$S(\{1, \ldots, n\})$, i.e. permutations of the $n$ vertices, but in the
case of the $1$-gon there is just one permutation, the identity, and in
the case of the $2$-gon there are just two (identity and one
transposition). Furthermore by a reflection of the $2$-gon in the
horizontal axis it gets projected onto itself (every vertex stays where it
is), so this reflection is actually the identity, and not a separate
element $r \ne 1$. Or does I miss something? Could it be consistent to
look at $D_1$ and $D_2$ in this way?