Is the term exogenous in OLS a misnomer? in a simple linear regression model 
$$y=\beta _1 + \beta _2 x + \epsilon$$
We define $x$ to be exogenous if 
$$E(\epsilon|x)=0$$
I am a bit puzzled as to why this term is called "exogenous", which I intuitively understand to mean something like "not causally influenced by the other variables in the model, including $\epsilon$, since it clearly does not mean that $\epsilon$ and $x$ are independent. We can have for example, that the variance of $\epsilon$ is a function of $x$, or that $\epsilon$ has a t-distribution with $x$ degrees of freedom, or something weird like that. 
So what is the intuitive reason that $x$ is called exogenous if that condition applies, rather than independence?
 A: You are right, and you are touching  the achilles heel of most econometrics textbooks: the conflation of causal and statistical concepts. $E[\epsilon|x] = 0$ , by itself, formally is just mean independence. And in fact, any regression error will be mean independent of the covariates. That is, any variable $Y$ can be decomposed as $Y = E[Y|X] + \epsilon$ where $E[\epsilon|X] =0$. 
Thus, the concept of exogeneity has to go beyond associational ideas, such as mean independence --- it makes sense in a structural framework, where the parameters you want to estimate mean something more than merely a description of the observed distribution. And your intuition about the causal content is on the right track. 
The general definition of exogeneity of $X$ is that $X$ is independent of  all other unobserved factors that causes $Y$ except those mediated by $X$ itself -- $\epsilon$ is just a shortcut to represent all those unmodeled factors. In a linear structural equation setting, however, this is usually relaxed to claiming only that the mean of $\epsilon$ is independent of $X$, which is weaker but enough to identify the structural coefficient $\beta$ in that setting.
