Let $\nabla$ denote a linear connection and for some curve $c:I\to M$ let $D_t$ denote the covariant derivative along $c$ with respect to $\nabla$. Some people define a geodesic to be a curve s.t. $D_t c \equiv 0$ whereas others write $\nabla_{c'} c' \equiv 0$. First notice that the latter one is a well defined expression since the connection does only depend on the first argument in the point and on the second on a curve that goes to the point with the appropriate velocity.
Now I want to understand why both definitions are equivalent.
In the times where $c$ is regular we can extend $c'$ to some neighborhood and hence by the properties of covariant derivative along a curve both expressions agree, but what is when $c'(t)=0$. Can we get it by some continuity argument?
The answer is that both are equivalent (almost) by definition.
$\nabla_{c'} c' \equiv 0$ has some problems. What if $c$ auto-intersects somewhere? The fact is: $c'$ need not be a vector field on $M$, nor induced by one. You have some different ways to handle this: you can consider local information (that is, you consider small neighbourhoods on $I$ and see $\nabla_{c'} c'$ there), or you can consider that $\nabla$ is not really $\nabla$, but really the pull-back connection induced on the pull-back of the tangent bundle by $c:I \to M$, and $$\nabla_{c'} c':=\nabla^*_{\partial/\partial t}(c^*(Tc)),$$ by definition. Any way you want, you must define what it means. And when you see the definition, you will see that they are equivalent by construction (either of the pull-back connection or of the covariant derivative, depending on how you do it).
More explicitly, if you follow the way doCarmo does it, your equivalence follows from his Proposition $2.2$.