Here are the two references I am looking at for this equation:
https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc432.htm
https://en.wikipedia.org/wiki/Exponential_smoothing
One says $$S_{t+1} = \alpha y_t + (1 - \alpha)S_t,$$ And the other says $$s_{t} = \alpha x_t + (1 - \alpha)s_{t-1}.$$
If you change the indices on any one of the equations to make it match the other, say for example if I change the second one I listed to match the first one:
$$s_{t+1} = \alpha x_{t+1} + (1 - \alpha)s_{t},$$
then change the corresponding letters to match (capitalizing the '$S$' and changing $x$ into $y$):
$$S_{t+1} = \alpha y_{t+1} + (1 - \alpha)S_{t},$$
you can see the two equations do not match with the first of the two terms on the right side.
I am not sure which one is the correct equation to learn since I am learning this on my own.