The total sum of squares = treatment sum

I am confusing Sum of Squares information.  Can you please tell me the difference between these two:

The total sum of squares = treatment sum of squares (SST) + sum of squares of the residual error (SSE)

Versus:

When computing F stat you use

Between group SS/(g − 1) Within group SS/(g(t − 1))  Isnt Within Group SS then effectively the SSE?

Sum of squares between groups or sum of squares treatments are actually identical. Different authors use different names to define different situations.

For example, suppose we want to test how the drug dosage affects the blood pressure, here we are using some outside treatment so sum of square treatment should be used.

Consider another example where we want to test the income between different races, here no outside treatment is used thus sum of square groups is more appropriate.

Again, for regression we use sum of square regression,

But all of them are representing the same identical quantity so it basically does not matter if you use sum of squares group or sum of squares treatment if you are confused.

Similarly,

SS within group = SS error = SS residual.