Assume that the distributions here are such that
$$E[y\mid (x,z)] = \beta_2 x+\beta_3 z$$
Then we can write
$$y = E[y\mid (x,z)] + e_{xz}$$
where $e_{xz}$ is the conditional expectation function error (of the specific conditional expectation). By design, as can be easily verified,
$$E[e_{xz} \mid (x,z)] = 0 \implies E[e_{xz} x] = E[e_{xz} z]=0$$
Assume further that it holds that
$$E[y\mid x] = \lambda_2 x$$
Here too we can write
$$y = E[y\mid x] + e_{x},\qquad E[e_{x} \mid x] = 0$$
In general, we expect that $\beta_2 \neq \lambda_2$.
So which is the "true" coefficient? Which coefficient expresses the "true" relationship?
Both and neither.
Both, because both $\beta_2$ and $\lambda_2$ reflect a valid statistical relationship between $y$ and $x$. In the first case, with $z$ present. In the second case, with $z$ absent.
Neither, because, the word "true" can only be associated with a causal relationship -and nowhere up to now did we argue about causal relationships, only about statistical relationships.
But econometrics is the arrogant child of statistics that believes it can argue about causal relationships all the time, when the parent frowns, or worse, every time the word "causality" is uttered.
This happens because econometrics is the applied arm of a social/behavioral science, economics. And economics starts by theorizing on causal relationships. In our example, assume that this theorizing leads to the specification using the $x,z$ variables. Then we are "forced" by our own theoretical/behavioral/causal arguments to consider the specification with the $(x,z)$ and the betas as the correct one.
So from the two valid statistical relationships (or from the "infinite" number of such valid statistical relationships), we want to estimate the first specification, because we consider it to be the "causally true" specification, because, we argue, all other effects on $y$ are unimportant, average to zero, and do not affect the causality carriers, $(x,z)$.
But estimators are like computers: they do what they tell them to do, not what they want them to do. If we give OLS the $(y,x)$ sample, we "tell" it, whether we like it or not, to estimate the lambda coefficient.
To see this, applying OLS we will obtain
$$\hat \beta_2 = \frac {\sum x_iy_i}{\sum x_i^2}$$
Now, we can substitute in this expression either one of the expressions for $y$.
So
$$\hat \beta_2 = \frac {\sum x_i(\beta_2x_i + \beta_3z_i+e_{xz})}{\sum x_i^2} = \beta_2 + \beta_3\frac {\sum x_iz_i}{\sum x_i^2} + \frac {\sum x_ie_{xz}}{\sum x_i^2} \to_p \beta_2 + \beta_3\frac {E(xz)}{E(x^2)}$$
But also,
$$\hat \beta_2 = \frac {\sum x_i(\lambda_2x_i + e_{x})}{\sum x_i^2} = \lambda_2 + \frac {\sum x_ie_{x}}{\sum x_i^2} \to_p \lambda_2$$
So no matter how we desire to "baptize" the estimator ("$\hat \beta_2$" in our case, indicating what we want it to estimate), it will estimate consistently $\lambda_2$: it does what we really told it to do, not what we wanted it to do.
The customary way to say this, is declaring that "the regressor is correlated with the error term", but here we are no longer referring just to the conditional expectation function error but to the "error"
$$u = \beta_3 z+ e_{xz}$$.
This is our way to say that we want to estimate the beta but we can only say to the estimator to estimate the lambda.
