I believe that your $\tau_g$ is what É. Cartan calls the 'geodesic torsion' in his 1945 book Les systémes différentiels extérieurs et leurs applications géométriques. He denotes his geodesic torsion by $1/T_g$. He gives your formula for $1/T_g$ in Chapter 7 as part of equation $(17)$.

The point is that what you call $\tau_g$ is simply the value of what Cartan calls the third fundamental form $\Psi$ of the surface $S$ evaluated on the tangent vector to the curve, which is why it depends only on the tangent vector and not on any higher derivatives. (Note: Cartan's 'third fundamental form' (see equation $(14)$ in Chapter 7) is not what is nowadays called the 'third fundamental form' and usually denoted $I\!I\!I$. The modern $I\!I\!I$ is just the pullback via the Gauss map of the standard metric on the $2$-sphere.)

I believe that your $\tau_g$ is what É. Cartan calls the 'geodesic torsion' in his 1945 book Les systèmes différentiels extérieurs et leurs applications géométriques. He denotes his geodesic torsion by $1/T_g$. He gives your formula for $1/T_g$ in Chapter 7 as part of equation $(17)$.

The point is that what the OP calls $\tau_g$ is simply the value of what Cartan calls the third fundamental form $\Psi$ of the surface $S$ evaluated on the tangent vector to the curve, which is why it depends only on the tangent vector to the curve and not on any higher derivatives.

Note: Cartan's third fundamental form $\Psi$ (see equation $(14)$ in Chapter 7) is not what is nowadays called the 'third fundamental form' and usually denoted $I\!I\!I$. (The modern $I\!I\!I$ is just the pullback via the Gauss map of the standard metric on the $2$-sphere.) Instead, if, in an orthonormal coframe field, the first fundamental form of $S$ is $I = {\omega_1}^2 + {\omega_2}^2$ and the (usual) second fundamental form is $I\!I = h_{11}\,{\omega_1}^2 + 2h_{12}\,\omega_1\omega_2 + h_{22}\,{\omega_2}^2$, Cartan's third fundamental form is $$ \Psi = h_{12}\,{\omega_1}^2 + (h_{22}{-} h_{11})\,\omega_1\omega_2 - h_{12}\,{\omega_2}^2. $$

I think that your $\tau_g$ is the reciprocal of what É. Cartan calls the 'geodesic torsion' in his 1945 book Les systémes différentiels extérieurs et leurs applications géométriques. He denotes his geodesic torsion by $T_g$. He gives your formula in Chapter 7, Section 2.

The point is that what you call $\tau_g$ is simply the value of the third fundamental form $I\!I\!I$ of the surface $S$ evaluated on the tangent vector to the curve, which is why it depends only on the tangent vector and not on any higher derivatives.

I believe that your $\tau_g$ is what É. Cartan calls the 'geodesic torsion' in his 1945 book Les systémes différentiels extérieurs et leurs applications géométriques. He denotes his geodesic torsion by $1/T_g$. He gives your formula for $1/T_g$ in Chapter 7 as part of equation $(17)$.

The point is that what you call $\tau_g$ is simply the value of what Cartan calls the third fundamental form $\Psi$ of the surface $S$ evaluated on the tangent vector to the curve, which is why it depends only on the tangent vector and not on any higher derivatives. (Note: Cartan's 'third fundamental form' (see equation $(14)$ in Chapter 7) is not what is nowadays called the 'third fundamental form' and usually denoted $I\!I\!I$. The modern $I\!I\!I$ is just the pullback via the Gauss map of the standard metric on the $2$-sphere.)