What defines a data generating process Let’s say we have a certain DGP $y=c+ax+bz+ error$. I take this to mean that conditional on the values of $x$ and $z$ in a certain instance (and no additional information )the expected value of $y$ can be inferred by plugging the values into the above equation. Now let’s say $x$ and $z$ are correlated. A different DGP may now be $y=d+mx+error$ where m may be higher(or lower) than $a$ due to the correlation of $x$ and $z$. Now wouldn’t $y=d+mx+error$ also be a valid DGP given that the expected value of $y$ conditional on $x$ (and no other information) is given correctly?
There seem to be (at least) two ways of interpreting the concept of a 'data generating process':


The data generating process is a correct description of the causal process
that generates the values of the dependent variable $Y$.
The data generating process specifies the expected value of the dependent variable $Y$ given the values taken the independent variables.


Now the answer to your question seems to depend on which interpretation we take:


If we follow interpretation 1, then these cannot be both data generating processes: either variable $Z$ causally determines $Y$ or it does not.
If we follow interpretation 2, then these can both be valid data generating processes. As you point out, there is no reason to suppose that $\mathbb{E}[Y|X = x]$ is the same as $\mathbb{E}[Y|X = x, Z = z]$.