Suppose that $G$ and $G'$ be the vector fields on $\mathrm{T}M$ that describe the first and the second definition of the geodesic flow; we need to show that $G=G'$.
If the metric tenor $g$ is flat, then $G=G'$ (it should be obvious).
Now assume that two metric tenors $g_0$ and $g_1$ on $M$ coincide at a point $p$ up to first order. Observe that $G_0=G_1$ and $G'_0=G'_1$ on $\mathrm{T}_p$.
Finally, observe that given a Riemannian metric $g_1$ there is a flat metric $g_0$ that coincides with $g_1$ up to first at a given point $p$.