What expresses if/how two variables are dependent on each other?
I think what the clarification has in mind is that correlation does not imply causation, that is, that two variables are mathematically correlated, does not mean they are dependent on each other. 

Here a wikipedia article if you'd like to expand on reading.

Here is a page full of examples where two variables, although highly correlated, are very clearly independent one from the other. For example: Per capita consumption of chicken vs Total US crude oil imports, or Per capita consumption of mozzarella cheese vs Civil eng. doctorates awarded.



Having shared my understanding of the clarification, now going back to your question:


  What is there that is related to if/how two variables are dependent on each other?


I think what does show if two variables are dependent on each other is not correlation alone, but a further analysis based on common sense. 



Hope this helps.