Derivation of the Geodesic Equation from Action Principle

Question

So there is this derivation here Geodesic Equation from Action Principle where the curvelength on some geometry is minimized. The derivation starts with some parameter $\lambda$ which gradually turns into the proper time $\tau$ -.-. Also at some point all of a sudden $\sqrt{-g_{\mu \nu} \, \frac{{\rm d}x^\mu}{{\rm d}\lambda} \frac{{\rm d}x^\nu}{{\rm d}\lambda}}$ is set to $1$ in the denominator, which is only legit if $\lambda=\tau$, but not for some arbitrary parameter $\lambda$. This is crucial, because if it would not be 1 (or constant), there would be another term arising after the partial integration which eventually leads to the equations of motion. So my question is, is the geodesic equation as typically given (top of the article with $s$) only valid if the velocity field $\frac{{\rm d}x^\mu}{{\rm d}\lambda}$ has unit (or constant) norm, but not for any arbitrary parameter $\lambda$?