The order of the identity element of a group is usually considered to be one because $1 * e = e$ or $e^1 = e$.
However, the text on torsion points (from Mathematical Cryptography by Silverman, Pipher etc) says the following
Let $m \ge 1$ be an integer. A point $P \in E$ satisfying $mP = \infty$ is called a point of order $m$ in the group $E$. We denote the set of points of order $m$ by $E[m] = \{ P ∈ E : [m]P = \infty\}$. Such points are called points of finite order or torsion points.
However, further down, he includes $\infty$ in the set of 2-torsion points. I think $\infty$ will be included in all different sets of n-torsion points - it will be in $E[2]$, $E[3]$ ... $E[n]$
Considering this, isn't it better to define it as "We denote the set of points in $E[m]$ to be those points for which satisfy $mP = \infty$" rather than defining it as "We denote the set of points of order $m$ by $E[m]$"?