There is a page from the book Algebra Interactive, by Arjeh Cohen, that states in Ex. 5.1.10 that the order of permutation $[2\,\,1\,\,3]$ is $2$.
The book states on the same page the defn. of 'Order' of a permutation as:
Order of a Permutation: The order of a permutation $g$ is the smallest positive
integer $m$ such that $g^m = e$.
On trying out: $[2\,\,1\,\,3]\,\,[2\,\,1\,\,3]$, get the mappings from right to left
$$2\rightarrow 1\rightarrow 3$$ $$1\rightarrow 3\rightarrow 2$$ $$3\rightarrow 2\rightarrow 1$$
However, on getting the mapping one more time, i.e. $m=3$, get
$$[2\,\,1\,\,3]\,\,[2\,\,1\,\,3]\,\,[2\,\,1\,\,3]:$$
$$2\rightarrow 1\rightarrow 3\rightarrow 2$$ $$1\rightarrow 3\rightarrow 2\rightarrow 1$$ $$3\rightarrow 2\rightarrow 1\rightarrow 3$$
So, is the book wrong in taking order as $2$.
Community wiki answer so the question can be marked as answered:
As already pointed out in comments, the list notation is being used here, where $[2\ 1\ 3]$ refers to the permutation that maps $1$ to $2$, $2$ to $1$ and $3$ to itself. Thus, this is indeed a transposition of order $2$.