Thank you, this is very high-quality feedback, with a lot of clear advice on how I can improve the post. I will do my best to make improvements over the next few days. I greatly appreciate that you took the time to draw my attention to these things. Among many other things, you have convinced me that a lot of things about the set-up are not obvious to non-epidemiologists. The article may need extensive restructuring to fix this.

You are obviously completely right about my abuse of the term "empirical research". I will fix it to something like "observational correlation studies" tomorrow.

So I'm trying to puzzle this out. What if the values for A,L,Y are binary, and in the diagram on the left, L->A and L->Y always just copy values from L to those other two deterministically; while in the diagram on the left, let's say that A is always 1, L is randomly chosen to be 0 or 1 (so that its dependence on A is vacuous), while Y is a copy of L. Then the joint distribution generated by graph 1 will be, in order ALY, 100 or 111 with equal probability, and it cannot be generated by graph 2, because in any distribution generated by graph 2, A=Y in all samples.

I agree that this part of the post needs more work. I think what is happening, is that you have data on a probability distribution that was generated by graph 1, and are then asking if it could have been generated by a particular mechanism that can be described by graph 2. However, the point I wanted to make is that you would have been able to come up with some mechanism described by graph 2 that could account for the data.. I realize this is not clear, and I will work on it over the next few days.

why oh why do you write a=0 and a=1 in the conditioned variables instead of A=0, A=1?

When I use lower case a, I am referring to a specific value that the random variable A can take. Obviously, I agree that I should have spelled this out. For example , the counterfactual Y(a) describes would have happened we intervened to set A to a, where a can be either 0 or 1. The distinction between upper case and lower case is necessary..