Solved – Using Holt-Winters formula, how do you choose which seasonality to begin your first forecast period with

Yes.

In Holt-Winters, you will initialize and update three different components, one for the level, one for the trend and one for the seasonality. The seasonality component actually has as many entries as there are periods in one seasonal cycle – 4 for quarterly data, 12 for monthly data etc.

When you forecast out, the forecast includes the most recent level component estimate, the most recent trend component estimate (extrapolated, possibly dampened) and the most recent seasonality component estimate for the seasonal period you are interested in. Thus, when forecasting for next April, you'd use the entry corresponding to April.

I recommend this section on Holt-Winters in Hyndman's and Athanasopoulos' excellent free online forecasting textbook. Note in particular which seasonal entry is used in forecasting: $s_{t-m+h_m^+}$, where $s$ contains the updated seasonal components, $t$ is the forecast time period, $m$ is the length of the seasonal cycle, and

$$ h_m^+ = lfloor (h-1) text{mod} mrfloor +1$$

for a forecast horizon $h$ ensures that you take the correct seasonal entry.