By the very definition of power you must use the effective value of an AC voltage for this calculation. Otherwise the definition of power was different for AC and DC, which doesn't make any sense.
In the case of a pure sine wave this effective value is the peak voltage divided by sqrt(2). For other wave forms the effective value would have to be calculated differently.
But as long as we are talking about a system within its limits of linearity,.other waveforms can always be created by the superposition of individual sine waves, anyway, so there is some universal truth in dealing with sine waves, only.
I don't want to introduce the peak to peak value into this discussion to not feed further confusion.