Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, \gamma(t_i)), \end{equation}
where the infimum is taken over all smooth paths $\gamma$ connecting $x$ to $y$ and the supremum is taking over all subdivisions $(t_0, \ldots, t_N)$ of $\gamma$.
The metric space formed by $(X, \widehat{d})$ is called a length space. However a problem I'm looking at refers to a length metric. My question is what is a length metric? Is it just a metric $d$ that is defined as above?