In the mathematical literature, the term "Cartesian coordinates" is used most frequently to refer simply to the standard coordinate functions on $\mathbb R^n$, namely the functions $x^1,\dots,x^n\colon \mathbb R^n\to \mathbb R$ defined by $x^i(a^1,\dots,a^n) = a^i$. Somewhat less frequently, I've also seen the term used to refer to any coordinate system on $\mathbb R^n$ obtained by composing the standard coordinates with a rigid motion, which can also be characterized as those coordinates for which the standard coordinate vectors $\partial/\partial x^1,\dots,\partial/\partial x^n$ are orthonormal.
The point is that it only makes sense to talk about "Cartesian coordinates" on $\mathbb R^n$ itself, or on an open subset of $\mathbb R^n$. On an arbitrary smooth manifold, the term has no meaning. Of course, on any smooth manifold $M$, each point has a neighborhood $U$ on which we can find a smooth coordinate chart, and such a chart allows us to identify each point $p\in U$ with its coordinate values $(x^1(p),\dots,x^n(p))\in\mathbb R^n$, and thus to temporarily identify $U$ with an open subset of $\mathbb R^n$; but we would not call these coordinates "Cartesian coordinates on $M$."
If your manifold $M$ is endowed with a Riemannian metric $g$, then there is more that can be said. For example, one could ask whether it's possible to find a coordinate chart in which the given Riemannian metric has the same coordinate expression as the Euclidean metric: $g= (dx^1)^2 + \dots + (dx^n)^2$. If this is the case, then geodesics and distances within this coordinate neighborhood are given by the same formulas as they are in Euclidean space; but that might not hold true elsewhere on the manifold. I think this might be the question you're getting at in your last paragraph, although I would not call these "Cartesian coordinates" because they don't have an open subset of $\mathbb R^n$ as their domain. Off the top of my head, I don't know of any standard nomenclature for such coordinates, but it wouldn't be inconsistent to call them "Euclidean coordinates" or "flat coordinates."
It's a basic theorem of Riemannian geometry that it is impossible to find such coordinates unless the curvature tensor of the Riemannian metric is identically zero on the open subset $U$. You'll find a proof of this fact in virtually any book on Riemannian geometry, such as my Riemannian Manifolds: An Introduction to Curvature (Theorem 7.3). If you want a treatment that doesn't use so much of the machinery of Riemannian manifolds, my Introduction to Smooth Manifolds has a proof that it's impossible to find Euclidean coordinates for the ordinary $2$-sphere in $\mathbb R^3$ (Proposition 13.19 and Corollary 13.20).
Best Answer
Euclidean space can have many meaning on what kind of space you consider, because you may have to define more or less things for the space to work. Usually we're speaking of the Euclidean vector space, over which you must define a sound operation $+$, $-$ and $\cdot$ that must obey the axioms of a vector space. Usually the easiest way is to use $\mathbb R^n$ and do the addition and scaling component wise for the tuple, but you could use any set with well-defined operations $+$, $-$ and $\cdot$ that is isomorphic to that. For instance, $\mathbb C$ with its standard definition of $+$, $-$ and $\cdot$ is a 2-dimentional Euclidean space.
You could also speak of the Euclidean affine space, which is made of a vector space and a set of point with no definite center. For example the real world (actual physical considerations apart) with the vector space $\mathbb R^3$ and an operation that convert the direction given by two points to a vector and another that translate a point along a vector. When informal and using the same underlying set for the vector space and the set of points, mathematicians may not distinguish points from vectors clearly.
A real vector space is synonymous to an Euclidean vector space, with the added convention that the underlying set is $\mathbb R^n$ for some $n$. A real affine space is an affine space build unpon a real vector space.
Cartesian coordinates, in a affine space, is a coordinate system over points using a orthogonal system. A point coordinate in the system is determined by the distance it has to each axis. Other systems include polar coordinates, if you have defined a dot product over your vector space, that uses angle and the distance to an "origin" point.
I've never heard of real coordinates. I guess it's synonym to cartesian coordinates or coordinates of a real (affine) space.