Definition and delimitation of regression model
In the past me too shared your perplexity about this point. You refers on econometrics literature and me too refers primarily on that. Unfortunately most econometrics books do not helped much. However I achieved a clearer view that seems me consistent.
What is the definition of a regression model?
It seems me that the correct definition of Regression is as synonym for Conditional Expectation Function (CEF); this definition come from statistical literature. Therefore we can realize that all depends from the joint distribution of the random variables involved $D(y,X)$. The regression is
eventually we can speak of regression in error form representation
read here for more about that: Regression and the CEF
Frequently people speak about linear regression (model), indeed this is the king object of econometrics. But from previous definition we can realize that linear regression is an explicit specification for $g(X)$. Moreover is possible to show that the true meaning of the famous mean independence assumption $E[\epsilon|X]=0$ is a restriction on $D(y,X)$; it imply linearity for the CEF.
Already here we can realize that assume both linearity for the regression and mean independence for his error is redundant. Worse, most econometrics books speak about Exogeneity assumption in place of mean independence assumption and attribute to it a crucial but strongly different meaning; meaning that cannot be attributed to a regression. Worse again, sometimes this assumption is given in the form $E[\epsilon X]=0$ that is always true by construction in all regressions! This fact reveal undoubtedly the contradiction.
Two book the you (Richard Hardy) cited in your reply, Wooldridge 2012 and Greene (2011), are among them.
The core of the problem swing around a conflation among statistical and causal concepts. Indeed exogeneity is, or should be, a causal concept and not an assumption about regression.
What is not a regression model?
A structural equation is not a regression equation (model). The conflation between the two concepts seems me the root of the problems in econometrics literature. Indeed when econometric authors speak about Exogeneity (in any form defined) have in mind something like a structural equation not a regression equation.
These my replies go deep about those point:
How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?
Under which assumptions a regression can be interpreted causally?
What is the relationship between minimizing prediction error versus parameter estimation error?
moreover other conflation is among the meaning of regression intended as theoretical quantity and properties of his estimator, primarily OLS estimator. For example the “No multicollinearity” assumption is frequently intended as a necessity for uniqueness of OLS estimator, it is an algebraic condition that deal with the data at hand and have few to share with the statistical properties of the random variables involved.
Finally, all above for to said (also) that I do not suggest the use of word "regression" as a meta-concept in the sense suggested by kjetil b halvorsen; Indeed I fear that conflation between regression and structural equation come from that. Indeed concept like conditional quantile function can be interesting but refers on that as quantile regression is (widespread) bad custom. Moreover model like GARCH have much to share with the concept of Skedastic function different from regression one. About model like ARIMA I said that: AR subcase are surely regressions; ARMA is a regression that include unobservable terms; ARIMA looks like a regression but the use of integrated series can bring ad hoc statistical issues.