In some textbooks I've read, it is said that an assumption for OLS to be unbiased in the standard cross-sectional model $y_i=\alpha + \beta \cdot x_i +\epsilon_i$, we can use the assumption $E(\epsilon_i|x_1,...x_n)=0$.

Do we need this, or is OLS already unbiased if just $E(\epsilon_ix_i)=0$ holds (in addition to the other required assumptions)?

If so, is there a particular reason why the textbooks would want to talk about the stronger assumption $E(\epsilon_i|x_1,...x_n)=0$?

In some textbooks I've read, it is said that an assumption for OLS to be unbiased in the standard cross-sectional model $y_i=\alpha + \beta \cdot x_i +\epsilon_i$, we can use the assumption $E(\epsilon_i|x_1,...x_n)=0$.

Do we need this, or is OLS already unbiased if just $E(\epsilon_i|x_i)=0$ holds (in addition to the other required assumptions)?

If so, is there a particular reason why the textbooks would want to talk about the stronger assumption $E(\epsilon_i|x_1,...x_n)=0$?