I don't think "angle of curvature" is a proper mathematical term; searching for it, it appears to just mean the change in direction from one end to the other, commonly measured in terms of the angle between perpendiculars. What they are doing here is using two chords (of what you are assuming to be a circle) to find the center of that (approximate) circle, and measuring the angle at the center.
But the key idea in such a measurement would be to decide what to consider the two ends, which I suppose is a practical matter of agreement among people in that field, and not something for math to decide.
In your first illustration, they seem to have chosen a point that is beyond the claw proper, and in the second, they arbitrarily chose not to go further. "Accuracy" is in the mind of the definer.