I don't think Pearl's diagrams are defined via do(). I think I disagree with that statement even if you can find Pearl making it.

Well, the author is dead, they say.

There are actually two separate causal models in Pearl's book: "causal Bayesian networks" (chapter 1), and "functional models" aka "non-parametric structural equation models" (chapter 7). These models are not the same, in fact functional models are a lot stronger logically (that is they make many more assumptions).

The first is defined via do(.), you can check the definition. The second can be defined either via a set of functions, or via a set of axioms. The two definitions are, I believe, equivalent. The axiomatic approach is valuable in statistics, where we often cannot exhibit the functions that make up the model, and must resort to enumerating assumptions. If you want to take the axiomatic approach you need a language stronger than do(.). In particular you need to be able to express counterfactual statements of the form "I have a headache. Would I have a headache had I taken an aspirin one hour ago?" Pearl's model in chapter 7 actually makes assumptions about counterfactuals like that. If you think talking about counterfactual worlds that don't actually exist is dubious, then you join a large chorus of folks who are critical of Pearl's functional models.

If you want to learn more about different kinds of causal models people look at, and the criticisms of models that make assumptions on counterfactuals, the following is a good read:

http://events.iq.harvard.edu/events/sites/iq.harvard.edu.events/files/wp100.pdf

Some folks claim that a model is not causal unless it assumes consistency, which is an axiom stating that if for a person u, we intervene on X and set it to a value x that naturally occurs in u, then for any Y in u, the value of Y given that intervention is equal to the value of Y in that same person had we not intervened on X at all. Or, concisely:

Y(x,u) = Y(u), if X(u) = x

or even more concisely:

Y(X) = Y

This assumption is actually counterfactual. Without this assumption it's not possible to do causal inference.