differential geometry - A reference frame is any coordinate system or just a set of Cartesian axes?

In Physics the idea of a reference frame is one important idea. In many texts I've seem, a reference frame is not defined explicitly, but rather there seems to be one implicit definition that a reference frame is just a certain cartesian coordinate system with a known origin.

In that case when we usually talk about two frames $S$ and $S'$ with $S'$ moving with velocity $v$ with respect to $S$ along some direction of the frame $S$ we mean that we have two sets of Cartesian coordinates $(x^1,\dots, x^n)$ and $(y^1,\dots,y^n)$ and that the coordinates are related in a time-dependent manner, for example

$$\begin{cases}y^1 = x^1 + vt, \\ y^i = x^i & \forall i \neq 1\end{cases}$$

On Wikipedia's page on the other hand we find this:

In physics, a frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at different moments of time. It may also refer to a set of axes used for such representation.

So that a reference frame may be a coordinate system (now, since we are not talking about axes, it could even be spherical or polar) or the axes themselves.

So what really is a reference frame? Is it just a set of cartesian axes in euclidean space $\mathbb{R}^n$? Or can it really be any set of coordinates like spherical and polar (or even another ones on more general manifolds)?

Also, how can we understand intuitively the idea of a reference frame and how this relates to the actual mathematical point of view?

EDIT: from a mathematical standpoint, a coordinate system on a subset $U$ of a smooth manifold $M$ is a homeomorphism $x: U\to \mathbb{R}^n$. The books lead me to believe that a reference frame would be equivalent to this idea. There are however, some problems in this approach:

1. 
   

   Books usually talk about reference frames just in $\mathbb{R}^n$ making the tacit assumption that the coordinates are cartesian and relating the frame with the axes. If the space is not $\mathbb{R}^n$, in truth, Cartesian coordinates aren't even possible and will problably be curved.

   
   


2. 
   

   Reference frames are present in Newtonian Mechanics, so it should be possible to define them without resorting to the notion of spacetime.

   
   


3. 
   
   

   Coordinate system are ways to assign tuples of numbers to points. But a reference frame can move around, something I think a coordinate system as defined in mathematics can't.

   
   

These three points are the at the core of my doubt. Reference frames shouldn't need anything spacetime related to be defined since they are present outside of relativity. Coordinate system as defined in mathematics cannot move around, so that reference frames shouldn't be synonimous to coordinate systems. And finally, if the space is not Euclidean, Cartesian axes are not possible.

So based on this, what's really a reference frame?

Answer

Let $M$ be your spacetime, a smooth manifold equipped with (pseudo) Riemannian metric (for example $\mathbb{R}^{(1,3)}$ for special relativity).

The set of reference frames is the frame bundle over $M$, usually denoted $FM$. Explicitly a frame at point $p$ in $M$ can be viewed as an ordered orthonormal basis (with respect to the the inner product defined by the metric) for the tangent space at $p$, $T_pM$.

For example, in metrics with Lorentzian signature in dimension 4, these frames are related by rotations in $\mathbb{R}^{(1,3)}$, aka Lorentz transformations, as expected.