I was going to put this material in the thread where we discussed this topic but I'm unable to locate that thread. Sorry for the unorthodox creation of a new thread for this. I just thought it was important to talk about. :)

I've been thinking hard about our discussion about the term "Locally Flat" [See? I do listen carefully to what you and most others say. :) ]. I looked this up in MTW and turned to page 217. Do you hold that MTW are describing locally flat coordinates or locally flat spacetime? Do you agree or object that MTW are correct and/or are better understood than "locally flat coordinates", and whatever Hillman called it, etc. If you read box 1.3 on page 20 then I think you'll agree that there is no difference between what I said and what Hillman said.

I also recall someone asking about the definiton of "Local" is defined in "Differential Geometry," Erwin Kreyszig, page 2

As far as I could tell, in the referenced pages MTW uses the terminology "locally lorentz frames". They also mentioned that said observer was in a "curved space-time.

If you read box 1.3 then you'd have seen that "Local Lorentz Geometry" is probably what we both have in mind and under that subsection (II) MTW give both a coordinate free language and one with a "language of coordinates." The former was what I'm used to thinking and that latter is what you seem to prefer. To me it appears to be two different descriptions of the exact same thing, i.e. Local Lorentz Geometry.

When GR claims that spacetime is 'locally flat' that means the metric can be made Minkowski and the Christoffels made vanish AT A POINT i.e. in appropriately chosen local coordinate system around that point.

My interpretation of the term 'local' is a statement that depends on a function and it's derivatives evaluated at the same local point, not on the function/derivative at different points or the function integrated over points etc.

For a free-fall/inertial observer, it is possible to turn the metric to Minkowski and nullify all Christoffels ON his worldline. That produces the physical local coordinate system used by the observer.

For an accelerated observer, exactly ON his worldline and along the whole wordline, it is possible to turn the metric in Minkowski by using the physical local coordinate system of the observer but it's not possible to nullify all the Christoffells in that system. Section 13.6 in MTW.

I believe there is another way to determine local flatness and that's to use parallel transport around a closed loop surounding region of spacetime. The displacement from bringing the transported vector around such a region is determined by the Riemann tensor, which always has a non-zero value at any given point where there is spacetime curvature. However we measure differences in parallel transported vectors and not the Riemann tensor. It is always possible to make the loop small enough so that any change in the transported vector is beyond the precission of the deivices used to make measurement.