So Vpk = 12.6*.707 = 8.91 V. Given this peak voltage, why is it different from that of the half wave peak voltage equation?

Your math is wrong.
If Vrms= 0.707Vpk, what is the formula for Vpk?

 

Look at the graph on post #11 again.
The peak voltage is higher than the rms voltage.
There is a reason why it is called peak voltage!

 

So Vpk = 12.6*.707 = 8.91 V. Given this peak voltage, why is it different from that of the half wave peak voltage equation?

No need to talk about half-wave or full-wave at this point.

There is a fundamental relationship between sine wave peak voltage (amplitude) and rms voltage.
Learn this first.