How to use the initial values computed for the Holt-Winters model to update the model to time t=n? I have followed the technique for determining the initial level, trend and seasonal components for the Holt-Winters model as detailed by Rob Hyndman on https://robjhyndman.com/hyndsight/hw-initialization/.  The notation at the top of page 24 in Forecasting With Exponential Smoothing (R. Hyndman, et al -  https://link.springer.com/content/pdf/10.1007/978-3-540-71918-2.pdf) suggests that for a time series of values starting at time t=1 (y1...yn) these initial values correspond to time t=0.  This is surprising to me (I would have expected that they correspond to time t=1, the beginning of the training data), since for a something like a straight line the level would appear to be the y-intercept of the training data, which then would best correspond to the first training data sample at t=1.
With this confusion on my part, I'm unclear about how, after determining these initial values and the best parameters (alpha,beta,gamma,phi), to then update the model to the end of the training data. Something like update(y1...yn) seems required, but if the initial values correspond to the first piece of data, then something like update(y2...yn) could make more sense.
Any guidance is much appreciated.

The initial level $\ell_0$ is indeed the level for $t=0$.
First off, in your reference, the statement

...compute a linear trend on the first ten observations against a time
variable $t = 1, \ldots, 10$. Then set $\ell_0$ to be the intercept of the
trend

is quite explicit as you have pointed out, too.
Also, all the models, e.g. the state space model on page 6 or the Holt-Winters model on p.16, have for $y_t$ a formula involving $x_{t-1}$ or $\ell_{t-1}$, thus, if $t$ starts at $1$, $x_s$ and $\ell_s$ start at $s=0$.
