FWIW, I don't think that low air pressure affects the odometer at all. A flat tire still has the same circumference as an inflated one. And high air pressures would need to be so high that the steel/rubber tire would have to physically stretch which I don't think you can readily do.
As others have said the flexing sidewall is what makes the difference. Ever see the diameter of a dragster tire change at RPM? And the effective diameter of a flat tire is rim+tread (little to no sidewall). Inflation pressure controls the middle range of values.

