> 1. Is Greene being sloppy? Should he actually have written: $E(y|X)=X\beta$? This is a "linearity assumption" that actually puts
> structure on the model.

In a sense, yes and no. In the one hand, yes, [given current modern causality research][1] he is a bit sloppy, but just like most econometrics textbooks are, in the sense that they do not make a clear distinction of causal and observational quantities, leading to common confusions like this very question. But, in the  other hand, no, this assumption is not sloppy and is indeed different from simply assuming $E(y|X)=X\beta$.

The crux of the matter here is the [difference between the conditional expectation, $E(y|X)$,  and the *structural* (causal) equation of $y$][2]. The linearity assumption in Greene is a *structural* assumption. Let's see a simple example. Imagine the structural equation is:

$$
y= \beta x + \gamma x^2 + \epsilon 
$$

Now let $E[\epsilon |x] = \delta x - \gamma x^2$. Then we would have:

$$
E[y|x] = \beta'x 
$$

where $\beta' = \beta + \delta$. Moreover, we can write $y = \beta'x + \epsilon'$ and we would have $E[\epsilon'|x] = 0$. This shows we can have a linear conditional expectation $E[y|x]$ which by definition is going to have an orthogonal disturbance, yet the structural equation would be nonlinear. 

> 2. Or do I have to accept that the linearity assumption does not put structure on the model but only defines an $\epsilon$, where the other
> assumptions will use that definition of $\epsilon$ to put structure on
> the model?

The linearity assumption does define an $\epsilon$, that is, $\epsilon := y - X\beta$ by definition, where $\epsilon$ represents all other causes of $y$ we chose not to model. The other assumptions are used either for **identification** of the structural parameters (for instance, the assumption of exogeneity of $\epsilon$) or for derivation of **statistical properties of the estimators** (for instance, the assumption of normality or homoskedasticity).


  [1]: https://stats.stackexchange.com/questions/249767/which-theories-of-causality-should-i-know/299090#299090
  [2]: https://stats.stackexchange.com/questions/60430/correlation-regression-and-causal-modeling/302486#302486