Let $f=f(x_1,x_2)$ be a $C^1$ scalar valued function of two variables at the point $\vec{x}_0$. We know the directional derivative of $f$ at $\vec{x}_0$ in the direction of $\vec{v}$ (unit vector) is given by $$ \lim_{h \to 0} \frac{f(\vec{x}_0+h\vec{v})-f(\vec{x}_0)}{h} = D_{\vec{v}}f(\vec{x}_0).$$
What can we say about the following limit: $$\lim_{h \to 0} \frac{f(\vec{x}_0+h\vec{v} + h^2 \vec{w})-f(\vec{x}_0)}{||h\vec{v} + h^2 \vec{w}||}$$ where $\vec{w}$ (unit vector, linearly independent from $\vec{v}$) is another direction ? Is this the same as $D_{\vec{v}}f(\vec{x}_0)$ ?
No. A simple observation is that the sign of $h$ in the denominator is lost in the second limit, so that the two-sided limit would be different between these "definitions".
A more subtle Answer would delve into the inadequacy of $C^1$ continuity to establish the equivalance of the path derivative in the second expression with the directional derivative in the first, even if $\vec w$ is orthogonal to $\vec v$, not merely linearly independent.