Geodesic equation and arclength parametrization

Let $(M,g)$ be a Riemannian manifold and $\gamma\subset M$ a curve in $M$. Suppose I would like to show that $\gamma$ satisfies the geodesic equations $$\ddot{x}^k=-\Gamma_{ij}^k\dot{x}^i\dot{x}^j.$$

If $\gamma$ is not parametrized by arc length, $\gamma$ may not satisfy the geodesic equations, as this example demonstrates: Why is $\gamma(t)=(0,t)$ a geodesic in the hyperbolic plane?

My question is why a geodesic, which is not parametrized by arc length, may not satisfy the geodesic equations (as in the above example).

Do Carmo derives the geodesic equations on page 62 and states that a curve in $M$ is a geodesic iff it satisfies the geodesic equation. Where does he assume that the curve is parametrized by arc length?

Let $\nabla$ be the Levi-Civita connection of $(M,g)$ and $c$ be a geodesic of $(M,g)$, then by definition, one has: $$\overline{\nabla}_{\dot{c}}\dot{c}=0,$$ where $\overline{\nabla}$ stands for $c^*\nabla$, then since $g$ is $\nabla$-parallel, the induced metric $\overline{g}:=c^*g$ is $\overline{\nabla}$-parallel, so that: $$\frac{\mathrm{d}}{\mathrm{d}t}\overline{g}(\dot{c},\dot{c})=2\overline{g}(\nabla_{\dot{c}}\dot{c},\dot{c})=0,$$ therefore $c$ is parametrized at constant speed that is proportionally to the arc length. I want to insist on this last point, a geodesic is not necessarily parametrized at speed $1$. This little discussion established that geodesic are not purely geometric objects.
The geodesic equation stated by Do Carmo is simply the coordinate-dependent version of $\overline{\nabla}_{\dot{c}}\dot{c}=0$. Indeed, by the very definition of the Christoffel $(2,1)$-tensor, one has $\nabla=\nabla^0+\Gamma$, where $\nabla^0$ is the flat connection of the given chart of $M$. Therefore, as we saw, being a solution of the geodesic equation implies being parametrized proportionally to the arc length, so no need to assume that the curve satisfies this condition.
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