I think intervention is the key idea missing from the above discussion of which of the the derivative function and the integrated function is the cause and which is the effect. In the signal generator example, voltage is a cause of current because we can intervene directly on the voltage. In the car example, acceleration is a cause of velocity because we can intervene directly on acceleration. This is not too helpful on its own, but maybe it will point the discussion in a useful direction.
I think intervention is the key idea missing from the above discussion of which of the the derivative function and the integrated function is the cause and which is the effect. In the signal generator example, voltage is a cause of current because we can intervene directly on the voltage. In the car example, acceleration is a cause of velocity because we can intervene directly on acceleration. This is not too helpful on its own, but maybe it will point the discussion in a useful direction.