So I read online that one of the assumptions of Gauss Markov Theorem is: $$E[\epsilon_i]=0$$However, we also know that one of the assumptions for linear regression is the zero conditional mean: $$E[\epsilon|X]=0$$ So I'm wondering, why the difference here? Do we have to further assume mean independence for the first case, i.e. $$E[\epsilon|X]=E[\epsilon ]$$ to arrive at the strict exogeneity criteria? Also, whats the benefit of making these 2 assumptions, as opposed to just assuming $E[\epsilon|X]=0$?
A related discussion is here: What's the difference between "mean independent" and independent? where carlos says "mean independence is not an assumption for linear regression" which further confuses me.