I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on a manifold with a metric defined on it (or indeed in general, without a metric). Am I correct in saying that the Cartesian coordinate system is a special kind of mapping which directly relates the intrinsic distance between two points on a manifold to the 'numerical' distance between their coordinates in $\mathbb{R}^{n}$. As, in general, a coordinate patch on a manifold will have a non-Euclidean geometry, although it will be possible to construct a one-to-one mapping such that these points can be labeled by coordinates in $\mathbb{R}^{n}$, it will not be possible to construct a map that preserves the intrinsic distance between two points in this patch such that it corresponds to the 'coordinate distance' between their corresponding coordinates in $\mathbb{R}^{n}$. In other words, although we will be able to construct a coordinate map, it will be impossible to construct a Cartesian coordinate map for this patch, which requires a Euclidean metric (apart from within a small neighbourhood around each point in this patch)?
N.B. By "Cartesian coordinate system" I mean the usual coordinate system used for $\mathbb{R}^{n}$ in which the geometry is flat (i.e. has a Euclidean metric)