I understand the intuition behind defining the curvature $k$ of a curve $\alpha :I\rightarrow \mathbb{R}^3$ as
$$\frac{\big(|\alpha '||\alpha ''|-|\alpha ' \cdot \alpha ''|\big)^{1/2}}{|\alpha '|^3}$$
since such a formula is the reciprocal of the radius of curvature, which can be derived as in here. However, I fail to have any intuition as to why the definition of the torsion $\tau$ is
$$\frac{(\alpha '\times \alpha '')\cdot \alpha '''}{|\alpha '\times \alpha ''|^2}$$
even after understanding that such definition implies that the torsion is zero if and only if the curve is planar (if all I knew about the formula for the curvature $k$ was that it is zero if and only if the curve is a line, I'd also be dissatisfied).
What further motivates the above definition of torsion? Why does that formula intuitively describe torsion?
Edit: to be somewhat clearer, my goal is to understand why, given that other formulas have the property of being zero if and only if the curve is planar, this particular expression is chosen as the definition of torsion. Similarly, I would like to understand how someone who has never studied the topic of differential geometry could naturally derive the formula above as an expression for torsion (analogously as how one could naturally derive the expression for curvature by means of the radius of curvature), since as of right now the expression feels alien to me, disconnected from what it's supposed to represent.