David:

The idea just came to me as I was writing. I have no idea where it came from. I do not recall reading it, but that is perhaps just testimony to my poor memory. I think I can safely say I did not get it from Spivak, nor from Riemann, because I have read them only recently. When I looked it up for you, I seem to recall a Wikipedia article that mentions the difference between Euclidean and Riemannian geometry and may perhaps lead you to a useful reference, but looking again just now I did not find it again.

Yes, I find it helpful as an example of a manifold that conceptually does not start out as a numerically coordinatised set, but only becomes one as a consequence of its "distance", considered as a function of point pairs. I would be interested to hear of other examples.

Christopher