Variance/Standard Deviation Confusion

I notice here that 'standard deviation' was used heavily in the narration, but in the formula itself what is meant to be SD actually indicates sample variance.

The formula for estimating population variance(pooled variance formula) had 'S2' in it: = (n-1)S2 + (n-1)S2 but the narrator kept indicationg 'S2' as the 'sample standard deviation' when clearly that was the symbol for sample variance. This threw me off. A google search showed the pooled variance formula uses 'S'(sample variance)  to estimate the population variance and not 'S2'(sample standard deviation).

Can this be clarified as it just makes the lessons inconsistent and hard to have 'closure'. Is it a case of using variance all through during calculations or deciding to use SD all throughout the calculations?

Isn't the formula suppose to be a guide or do we have some sort of leeway to choose between either Var or SD?

Thanks.

PS: The calculations used SD and multiplied it by 2(implying sample variance).