Some questions about the Generalized Method of Moments

Here are three (somewhat related) questions about the (Generalized) Method of Moments. I have only just today started studying this method.

1. Concerning the following statement by Greene:

Why is $g$ not a function only of $Y_t$? My understanding was that $g$ should be computable from the sample, but if the true parameter (which is unknown) influences $g$, then we cannot compute it.

1. Greene states that for the Methods of Moments estimator, we have $E(m_k(y_l))=\mu_k(\theta_1,...,\theta_k)$, where I understand $m_k(\cdot)$ to be the $k$'th sample-moment and $\mu_k$ the $k$'th population moment.

However, Greene then states that for the Generalized methods of moments estimator we need the "moment conditions":

perhaps these $m_{l,K}$ refer to something other than the moments, but I don't understand why this equation is different for the Generalized method, compared to the (non-generalized) method of moments. (Greene doesn't seem to explain this, if I'm not mistaken).

1. This also partially justifies the following question: Is the "moment condition" synonymous with "moment equation"? Greene uses both words, without clearly defining a difference.

A trivial example is the MOM estimator of the parameter of an Exponential distribution; we need to know how the parameter relates to the population mean (a very simple $g(y,\theta_0)$), as in, $\mathbb{E}[Y_t |\theta_0] = 1/\theta_0$. We can then use the sample mean and set $\hat{\theta}_0 = 1/\bar{y}$.
1. The $m_{l,K}$ don't have to be the usual central moments, they are "generalized" in the sense of being somewhat arbitrary functions that have expected values rather than just integer powers of $y$ that have expected values (subject to various conditions, of course, that Greene outlines.) For example, for a weighted estimator, you might have $\mathbb{E}[y_t w(y_t)] = 0$, where the $w(y_t)$ downweight extreme values of $y_t$ for robustness purposes. For more examples, see the Scope section of the Wikipedia page.