What exactly is a "true" population model in linear regression? What do we mean by a true population model when talking about linear regression? Say I want to study the effects of years of schooling $S$ on wages. I posit the following two models:
$log(wage)=β_0+β_1S+ε$
$log(wage)=v_0+v_1S+v_2A+v$
where $A$ is ability. Which is the "true" population model and what exactly are the requirements for a model to be a "true" model? I know that in this case $v1=β1$ because both measure the same partial effects of S on log(wages).

*

*Must the "true" population model satisfy the classical regression model assumptions (i.e. zero conditional mean $E[ε|S,A]=0$ etc?

*Must the true model simply posit the "correct" functional form relationship between the variables of interest? (i.e. that the true population model must be linear or non-linear)

*If $log(wage)=β_0+β_1S+ε$ is the "true" population model, if I split $β_1=l_1+g_1$ and rewrite the "true" population model as follows: $log(wage)=β_0+l_1S+e$ where $e=ε+g_1S$. Is this new model also a "true" population model?
I ask because many econometrics textbooks (especially Wooldridge) introduce a regression model and say that it is a "true" model but never clearly define what exactly constitutes a "true" population regression model.


Words like "model" and "population" are often used very differently by different authors. Here is a description of a pretty standard usage:
We presume that there is a data generation process that creates all your data rows $(wage, S, A)$ and that those triplets satisfy:
$$
wage = f(S, A, \varepsilon),
$$
where $\varepsilon$ is some error random variable. Then, this $f$ describes the "true" model. Usually, we don't know $f$, but we often think we can justify presuming that this $f$ can be approximated by a linear function such as yours:
$$
wage = v_0 + v_1 S + v_2 A + \varepsilon .
$$
Then, once we have convinced ourselves that the true model is linear, we are then interested in approximating the "true" parameters $v_i$.
Your first question: Yes, this is presumed, otherwise the standard algorithms don't work anymore. Note that the true model would still be the true model if it didn't satisfy those properties, we just hope that it does for convenience.
Your second question: Sometimes, the word model only refers to the functional form and includes all possible parameter settings. But sometimes authors describe the same functional form but with different parameters as different models. This has to be understood from the context.
Your third question: Your second formula has an error term $e$ that is not anymore independent of the variable $S$ which then violates those standard rules that make the application of standard methods possible.
