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I am generally pretty sloppy in my terminology on this point and use "reference frame" almost synonymously with "coordinate system". Is this a correct distinction between them:
A coordinate system is a mapping from points in R4 to events in the manifold
A reference frame is an orthonormal basis in the tangent space at some event
A coordinate system is a mapping from points in R4 to events in the manifold
A reference frame is an orthonormal basis in the tangent space at some event