Osculating plane as limit of planes containing three points of curve
I'm sure the two definitions are equivalent if $\alpha''$ is continuous in a nighbourhood of $s_0$. One should verify of course.
Coming back to your main question, you can even forget my comments because from $f'(\xi_1)=0$ and $f''(\eta)=0$ you get directly (!) $\,\vec n=\alpha'(\xi_1) \times \alpha''(\eta)\,$ up to a constant factor so that the limit is $\alpha'(s_0) \times \alpha''(s_0)$ by continuity.
Note that, if $\,\alpha'(s_0) \times \alpha''(s_0) \neq 0$, then $\alpha'\times \alpha'' \neq 0$ in a neighbourhood of $s_0$ by continuity.