One way to reinvent curvature of a plane curve is as follows: Given a smooth, oriented curve $C \subset \Bbb E^2$---here $\Bbb E^2$ denotes $\Bbb R^2$ equipped with its usual Euclidean structure---pick an arc length parametrization $\gamma : I \to \Bbb R^2$ of $C$.
We can immediately define a preferred oriented, orthonormal frame $\mathcal E := ({\bf e}_1(s), {\bf e}_2(s))$ determined by the intrinsic geometry of the curve: Set ${\bf e}_1(s) := \gamma'(s)$ and take ${\bf e}_2(s)$ to be the vector given by rotating ${\bf e}_1(s)$ anticlockwise by $\frac\pi2$.
Critically $\mathcal E$ is equivariant under the group $\operatorname{SE}(2)$ of Euclidean motions of $\Bbb R^2$: If we pick $g \in \operatorname{SE}(2)$ and replace $\gamma$ with $g \cdot \gamma$, then ${\bf e}_i(s) \in T_{\gamma(t)} \Bbb R^2$ is replaced by $g \cdot {\bf e}_i(s) \in T_{g \gamma(s)} \Bbb R^2$, $i = 1, 2$. Since ${\bf e}_1(s)$ has unit length, ${\bf e}_1'(s) \perp {\bf e}_1(s)$, hence $${\bf e}_1'(s) = \kappa(s) {\bf e}_2(s)$$ for some function $\kappa$, the (signed) curvature of $C$, and since $\mathcal E$ is equivariant, so is ${\bf e_1}'(s)$ and hence so is $\kappa(s)$, which is thus an invariant under Euclidean motions.
To illustrate the general approach, it might be useful to see how you'd construct a curvature invariant for a curve $C$ in some other (homogeneous) geometry on the plane, e.g., special affine geometry. In this context we denoye the plane by $\Bbb A^2$. In this setting, the analog of an orthonormal frame is a unimodular frame, that is a frame $\mathcal A := ({\bf e}_1(t), {\bf e}_2(t))$ such that $$\det \pmatrix{{\bf e}_1(t) & {\bf e}_2(t)} = 1,$$ and the replacement for the group of Euclidean motions is the special affine group, $G := \Bbb R^2 \rtimes \operatorname{SL}_2(\Bbb R)$.
A natural first attempt at construction $\mathcal A$ would be to choose a parametrization $\gamma$ of $C$ and choose the frame $(\gamma'(t), \gamma''(t))$. (Notice that unlike in the Euclidean case, the first element of the frame does not determine the second, essentially because the group of motions is larger in this setting.) In order for this pair to comprise a frame along $C$, $\gamma'(t), \gamma''(t)$ must be linearly independent---this nondegeneracy condition is the affine analogue of the regularity condition for curves in Euclidean space, and we'll henceforth assume that the curve satisfies it. In particular, that implies that $\det \pmatrix{\gamma' & \gamma''}$ has the same sign everywhere, and for expository convenience we'll take it to be positive; the negative case can be handled similarly. This choice of frame is not necessarily unimodular, but notice that if we reparameterize $\gamma$ (via a standard abuse of notation) $\gamma(s) := \gamma(t(s))$ for some invertible $t$, then
$$
\frac{d \gamma}{ds } = t'(s) \frac{d \gamma}{dt }, \qquad
\frac{d^2\gamma}{ds^2} = t'(s)^2 \frac{d^2 \gamma}{dt^2} \,\,\,\,\left(\!\!\!\!\!\!\mod {\frac{d\gamma}{ds}}\right) ,
$$
hence $\det \pmatrix{\gamma'(s) & \gamma''(s)} = t'(s)^3 \det \pmatrix{\gamma'(t) & \gamma''(t)}$. This in turn suggests that we define the (special) affine arc length of $\gamma(t)$ to be
$$s(t) := \int_{t_0}^t \sqrt[3]{\det \pmatrix{\gamma'(t) & \gamma''(t)}} \,dt ,$$ an affine analogue of Euclidean arc length. We can parametrize any curve by special affine arc length, and by construction, a curve $\gamma(s)$ is parameterized by special affine arc length iff the frame $(\gamma'(s), \gamma''(s))$ is unimodular. This frame is canonical in the sense that it is equivariant under the action of $G$. It then follows from unimodularity that $\gamma'''(s) = \kappa(s) \gamma'(s)$ for some function $\kappa(s)$, the (special) affine curvature, and $\kappa$ turns out to be a complete invariant for nondegenerate curves in $\Bbb A^2$. (N.b. I think the formula at the link uses the opposite sign convention for $\kappa(s)$.) The curves with $\kappa = 0$ are the parabolas, and the curves with constant $\kappa$ are the conic sections.
For (much) more, see, e.g., Clelland's excellent From Frenet to Cartan: The Method of Moving Frames; the above discussion of invariants of curves in $\Bbb A^2$ is essentially an outline of solutions to Exercises 6.2.12–13 there.
