Hello Laura,
The projection model is a special case of the regression model. So you can get results from a projection model using -regress-.
The regression model assumes that the unobservables are, on average, unrelated to the regressors or any function of the regressors. For example, X^3, sin(X), cos(X), etc. The projection model concerns itself with linear combinations of the regressors. In that sense it is a special case. Your intuition about linear regression can be formed in terms of the projection model.
The literature of instrumental variables for linear models and, to some extent, GMM write the problem in terms of the projection model, where E(ex)=0 is called a moment condition. This is why I decided to present endogeneity in terms of the projection model.
All that being said, I think the post would have benefited from a conceptual discussion of the projection model. In other word, thanks for the question.