Very slightly different things happen when you compress the wheel without rolling it as shown in figure 2, compared to when you roll the wheel along the road.
Your figure 2 is correct for static loading if the tyre does not slip at all on the road. This is what would apply in the case of my static experiment described in the second post of this thread.
A feature of the this static compression is that the portion of the tyre which touches the ground first is not compressed at all in the longitudinal direction, whereas as we move towards the ends of the chord there has to be an increasing compression to fit in the tyre material which is at a small angle to the ground. The compression of the tyre thus varies along the length of the chord. This is OK for a static compression with no slipping.
In rolling something subtly different happens for the no slip case:
As the wheel rotates each new bit of tyre does not come into contact with the ground immediately underneath the axle - the touching point shown in your fig 2. It first touches at the end of the radius marked R in your fig 1:
As each new bit of tyre gets fed in to the contact patch it is compressed to fit. The longitudinal compression (which is all we are concerned with in this simplified argument) is cos(alpha). The compression remains constant as this bit of tyre passes on under the axle until it is released as the tyre lifts from the road at the end of the chord marked c in fig 1.
Whoops! Just as I have written this I see Stu, ever on the ball, has just posted. I have not yet quite worked out the implications of his post.