Consider not the abstract situation of B = dA/dt, but the concrete example of the signal generator. It would be a perverse reading of the word “cause” to say that the voltage does not cause the current. You can make the current be anything you like by suitably manipulating the voltage.
But you can make a similar statement for just about any situation where B = dA/dt, so I think it’s useful to talk about the abstract case.
For example, you can make a car’s velocity anything you like by suitably manipulating its position. Would you then say that the car’s position “causes” its velocity? That seems awkward at best. You can control the car’s acceleration by manipulating its velocity, but to say “velocity causes acceleration” actually sounds backwards.
But let this not degenerate into an argument about the “real” meaning of “cause”. Consider instead what is being said about the systems studied by the authors referenced in the post.
But isn’t this really the whole argument? If the authors implied that every relationship between two functions implies correlation between their raw values, then that is, I think, self-evidently wrong. The question then, is do we imply correlation when we refer to causation? I think the answer is generally “yes”.
Maybe I’m not quite understanding, but it seems to me that your argument relies on a rather broad definition of “causality”. B may be dependent on A, but to say that A “causes” B seems to ignore some important connotations of the concept.
I think what bugs me about it is that “causality” implies a directness of the dependency between the two events. At first glance, this example seems like a direct relationship. But I would argue that B is not caused by A alone, but by both A’s current and previous states. If you were to transform A so that a given B depended directly on a given A’, I think you would indeed see a correlation.
I realize that I’m kind of arguing in a circle here; what I’m ultimately saying is that the term “cause” ought to imply correlation, because that is more useful to us than a synonym for “determine”, and because that is more in line (to my mind, at least) with the generally accepted connotations of the word.