observing (literally seeing) coincidences between those events and local features of the reference body.

This distinction between observer and the observer's "apparatus" like coordinate systems, measurement tools etc. was dropped by many later writers, and today it is common to find the term "observer" used to imply an observer's associated coordinate system (usually assumed to be a coordinate lattice constructed from an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector (a frame field),