Agreed. But there must be some slip for the motorbike formula to apply. I now think that if there is no slip the overall geometry changes slightly to accommodate a shortened contact patch length. At the moment I struggling to see exactly what changes.
It seems to mean that there will slight difference of height of the axle when rolling on a sticky surface compared to just loading the static bike tyre as I did in my experiment to measure the deflection. However, I think my static measurement is still valid in demonstrating that the motor bike formula does not apply to my bike. I am still calculating...
You have demonstrated a geometrical way of deriving the motorbike formula. But this does not seem to be the whole story, since there must be slip over some portion of the contact patch to fit the tyre into the chord (or else the geometry of the simple diagram changes at the ends of the chord). I think it would be interesting to track down the actual derivation of the formula, perhaps by reading the rest of the pages not shown in the limited google book view. It would help to see if they have used anything further than the geometry you have used.