> But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ?

It *may* correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$...

... But it can just as well be $Y \rightarrow X, Y \rightarrow Z$.

Statistical models are present wherever there are causal models, but those statistical models are not *equivalent/identical* to the causal models themselves. 

A statistical model only describes the correlation, and it does not (need to) describe the causation. You can describe and fit statistical models without a description of an underlying causal model. 

Or at least, certainly the statistical model alone does not tell you anything about the causation (except that there is *some* underlying causal mechanism, but we do not know not which). In this sense it is not equivalent to a causal model. 

You could see a statistical model as the shadow of a causal model.