Actually, there might not be a good way to model or describe the difference between causal inference, correlation and causality.
Causality involves a deep understanding of a phenomenon in science.
For example, the standard model of physics is pretty good at describing the real world in a good enough manner because we understand a lot of it. The difference with correlation and causality, in my view, is human, scientific understanding of what things are. Formulas, data or drawing are not enough.
For example there might never be a way to prove natural selection, even if there is a lot of data available, but a lot of scientific consensus is enough to describe causality.
Ie., ML makes the assumption that data points are IID.
The whole purpose of science is to produce models which explain why data isnt IID.