I'm in trouble with the definition of reference system in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard Euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

I'm in trouble with the definition of reference system in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard Euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

Edit: as adviced in the comments I'd ask for feedbacks on the following.

Let's take Schwarzschild spacetime for instance. To derive the metric tensor field $g$ as solution of the Einstein Field Equations (EFE), one begins by assuming spherical symmetry for spacelike hypersurfaces of constant coordinate time $t$. Therefore one picks spherical coordinates $(r,\theta, \phi)$ although at this stage doesn't know yet what will be the physical interpretation of the radial coordinate $r$. Basically at the very beginning one assumes that, whatever will be the interpretation of $r$, the metric $g$ will be the same at all events on each spacelike hypersurface sharing the same $r$ value.

I'm in trouble with the definition of reference frame in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

I'm in trouble with the definition of reference system in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard Euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

I'm in trouble with the definition of reference frame in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a smooth map from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

I'm in trouble with the definition of reference frame in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).