Thanks for the comment! I am not confident in answering whether your summary is correct or not, partly because it looks like we come from different backgrounds which use different languages to describe the same things.

The point I am trying to make is that the joint distribution of A, B and C only consists of information such as "in 15% of the population, A=1, B=1 and C=1, in 5% of the population A=0, B=1 and C=1" etc

If the data was generated by a joint distribution, this seems like it would be something like an algorithm that just says "Assign 'A=1, B=1, C=1' with probability 0.15 and assign 'A=0, B=1, C=1' with probability 0.05" etc

However, for causal inference, it is necessary to model the world as if the joint distribution is generated by three separate algorithms: One for A, one for B and one for C. There are many possible sets of such algorithms that will result in the same joint distribution.

We will therefore need to set up the problem so that we explicitly state the order in which the variables are generated, and stipulate that the input can consist only of variables from the past.