I believe the best way to sum up the idea of observers used in almost all treatments of standard Special Relativity is what Schutz says in his General Relativity book: [...]

I can accept that this is a correct quantitative description of presently available treatments. But I cannot accept the characterization of observers as it is suggested by Schutz (and others), because it appears to deny what to me is an indispensible element of discourse of (the geometric-kinematic part of) Einstein's theory of relativity, and Special Relativity in particular.

Namely, to consider individual identifiable observers attached to (or even identified as) individual identifiable material points, as repeatedly and consistently described by Einstein himself (e.g. here and here):

• each capable of observing and identifying and recognizing others, and in turn of being observed and recognized;

• each capable of preserving the collected observations in memory and of determining which observations he, she, or it had collected in coincidence, or in which order;

at least in principle, for the purpose of thought-experimental description and understanding; and more or less even in practice.

On the other hand, in General Relativity things are different.

Apparently as represented by Schutz et al.; but certainly not for the notion of observer as individual capable of collecting and ordering observations.

[...] together with an orthonormal basis

Which foremost raises the question how an individual observer ought to determine such a basis in the first place.

[One observer] cannot describe worldline of particles or other observers, nor anything of the sort. [...]

Surely any one observer can (be thought of) observing and recognizing others having observed his/her/its own (signal) indications; and consequently, signal by signal, determine whose corresponding ping echos had been received back in coincidence, or in which order, or "not yet at all". This capability is ascribed to observers already in Einstein's initial presentation of SR, 1905.

From the interelations between such determinations of individual observers follow descriptions of their collective geometric relations between each other; such as "ping-coincidence lattices" described here.

[...] what gives meaning to the coordinates.

The possible sprinkling of events (or likewise: of selected observers, and their individual ordered sets of indications) with coordinate tuples, for the purpose of representing the geometric relations between events (or likewise: for representing the frame relations between the selected observers) through the "natural" topological or even metrical properties of real number tuples, is of course only subsequent and secondary to the determination of the geometric relations under consideration.

I believe the best way to sum up the idea of observers used in almost all treatments of standard Special Relativity is what Schutz says in his General Relativity book: [...]

I can accept that this is a correct quantitative description of presently available treatments. But I cannot accept the characterization of observers as it is suggested by Schutz (and others), because it appears to deny what to me is an indispensible element of discourse of (the geometric-kinematic part of) Einstein's theory of relativity, and Special Relativity in particular.

Namely, to consider individual identifiable observers attached to (or even identified as) individual identifiable material points, as repeatedly and consistently described by Einstein himself (e.g. here and here):

• each capable of observing and identifying and recognizing others, and in turn of being observed and recognized;

• each capable of preserving the collected observations in memory and of determining which observations he, she, or it had collected in coincidence, or in which order;

at least in principle, for the purpose of thought-experimental description and understanding; and more or less even in practice.

On the other hand, in General Relativity things are different.

Apparently as represented by Schutz et al.; but certainly not for the notion of observer as individual capable of collecting and ordering observations.

[...] together with an orthonormal basis

Which foremost raises the question how an individual observer ought to determine such a basis in the first place.

[One observer] cannot describe worldline of particles or other observers, nor anything of the sort. [...]

Surely any one observer can (be thought of) observing and recognizing others having observed his/her/its own (signal) indications; and consequently, signal by signal, determine whose corresponding ping echos had been received back in coincidence, or in which order, or "not yet at all". This capability is ascribed to observers already in Einstein's initial presentation of SR, 1905.

From the interelations between such determinations of individual observers follow descriptions of their collective geometric relations between each other; such as "ping-coincidence lattices" described here.

[...] what gives meaning to the coordinates.

The possible sprinkling of events (or likewise: of selected observers, and their individual ordered sets of indications) with coordinate tuples, for the purpose of representing the geometric relations between events (or likewise: for representing the frame relations between the selected observers) through the "natural" topological or even metrical properties of real number tuples, is of course only subsequent and secondary to the determination of the geometric relations under consideration.