Are all statistical models also causal models? I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus.
So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).
So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$  (Z is a confounder of X and Y, but X also partially causes Y through M).

If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.
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 $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?
edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear
edit #2:
If I write a model $Y + aX + bZ + e$ in a programming language, I could do so as a function, e.g. in Python
 def model(a,X,c,Z):
     return a*X + b*Z + np.random.randn()

So if I change the input $X$, it will cause the output of $model(...)$ to change, but I cannot do the opposite. Isn't that a causal model?


Are all statistical models also causal models?

No, they aren't.

So the point between Pearl's causal graphs and
rules for manipulating causal graphs appears to be to turn a causal
graph into a statistical model (e.g. a linear regression).

This statement, at best, is confusing. The main point of Pearl works is to separate clearly: statistical concepts vs causal concepts. Your question itself reveal that you do not grasped this, not yet. In fact statements like the follow

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 $X→Y,Z→Y$ ? If so, then is the Pearl method just finding a
transform of a causal graph into another causal graph that is easier
to work with or represent as a regression?

confirm my impression, definitely.
Apart that a regression equation remain a respectable math equation (too much general concept) causal graph do not stay for represent some regression. Worse, the (causal) concept of manipulation cannot apply to regression. Indeed causal graphs represent structural equations and them must be clearly separated from regression  equations. Indeed Pearl underscore repeatedly the opportunity to use different notations.

If I write a model Y+aX+bZ+e in a programming language, I could do so
as a function, e.g. in Python

It looks like a causal model. It is so because the definitions demanded from programming language imply a kind of equal sign that is definitional ($:=$), it is different from the standard equal sign ($=$). The standard equal sign is logically symmetric, the other is not and precisely for this reason it is what a structural equation (causal concept) need.
Said that, the main link between causal and statistical model is about identification (read here:Why do we need identification in causal inference?). For identification purpose Pearl suggest do-calculus (read here:What's the purpose of do-calculus?)
For more details read here:
Under which assumptions a regression can be interpreted causally?
Criticism of Pearl's theory of causality
