Jon, +1 for the great explanation.
If I may emphasize your last point: The whole reason for measuring AC voltage as an RMS value is to provide the simplest and best indication of the power it will deliver to a resistive load. The reason why the RMS calculation squares the voltage is that the power is equal to E^2/R. (See the square there?)
During the AC cycle, as the voltage rises from zero, the instantaneous power delivered to the load increases as the square of the voltage. The power falls to zero again at the next crossover point.
So, what happens during the negative half-cycle? The power rises again to the same peak value, because the square of a negative number is a positive number.
In fact, if you plot the square of a sine-wave voltage, you get another sine wave of twice the frequency! The lower peaks of the squared waveform "sit" on the zero line, and the upper peaks are at Epk^2, where Epk is the peak voltage.
Since the new double-frequency sine wave represents power delivered to the load, the average power is at the midpoint, which is (Epk^2)/2.
The voltage that represents that average power level is the square root of that level, which is sqrt((Epk^2)/2), which reduces to Epk/sqrt(2). Look familiar?
