Again, we're talking about very small differences, but, when a couple of engineers start talking about this sort of thing, the methods of analysis become far more significant than the magnitudes of the parameters involved.
Yes, loading and tire pressure affect the rolling circumference of a tire. This is why I said the measured circumference of a tire on the tire changing machine is "essentially the same" as it is in operation. As Barry points out, that "essentially the same" is expressed as a 3% difference. The strains involved in load and air pressure would also affect the rolling circumference. This is why I said that the rollout results with 10 and 50 psi would not be "EXACTLY" the same.
But, the point I was trying to get across is that these differences are NOT the result of the very obvious flattening of the tire at the footprint. Again, the 10 and 50 psi rollout tests would clearly show this. While I would expect a small difference in results (from the sources discussed in the preceding paragraph), it would be nowhere near the difference expected if the rolling radius was assumed to be the vertical distance from roadway to axle centerline.
This vertical distance from roadway to axle centerline comes into play, however, when calculating acceleration. Barry mentions the gearing effect of a change in rolling circumference ("However, a change in rolling circumference of a tire acts in a similar way as a change in final drive gearing."), but this is a small effect compared to the deformation of the tire at the footprint. In other words, the radius change from the change in rolling circumference could never account for the deflection at the footprint.
A common error...yes, even in "the industry"...is to use the wheel revolutions per mile data from the proving grounds and then calculate the radius for use in acceleration calculations. This is commonly found in SAE papers. Well, I shouldn't call this an "error." We're talking about such small differences that the familiar term "accurate within engineering accuracy" is more than sufficient.