When to factor duty cycle into power consumption calculations?
In your first calculation you are computing the average voltage when in fact this should be the RMS voltage. In your second calculation you are correctly calculating the power (one half cycle) and dividing by two to correctly get average power.

So, taking the average power of 50 watts and multiplying by 1 ohm then taking the square root delivers RMS voltage because: -

$$V = \sqrt{P.R} = 7.071 V$$

But you can calculate RMS voltage by squaring each voltage, taking the mean then square rooting: -

$$\sqrt{\dfrac{10^2 + 0^2}{2}} = 7.071V$$

If the duty cycle were 25% then it would be: -

$$\sqrt{\dfrac{10^2 + 0^2 + 0^2 + 0^2}{4}} = 5V$$

Notice how I've split the latter solution into 4 distinct time slots.