In linear regression, the assumptions are linearity, independence, (sometimes) normality, (sometimes) homoscedasticity.
But when people talk about independence, sometimes they say we need independent observations (e.g. https://math.stackexchange.com/questions/1530571/linear-regression-model-assumptions) and sometimes they say we need independent error terms (e.g. What is a complete list of the usual assumptions for linear regression? )
My question is, are these two somehow equivalent? I guess that is the case if we do not treat $x$ as random variables, but is this the correct connection?
