Brief Answer:
Neurologist or cardiologist, best someone with subspecialty in dysautonomia

Detailed Answer:
I read your question carefully and I understand your concern.

Those symptoms you describe may be caused by POTS but they are not specific, could be caused by many other issues. So diagnosis can't be made on those symptoms alone. I would have to take your cardiologist's word for it, assuming he has made the necessary tests like prolonged ECG, ultrasound, tilt table testing etc in order to confirm that diagnosis.

As for who should treat it, if it is a case of POTS I would say that both neurologists and cardiologists would be equally qualified to deal with it. Problem is that autonomic disorders, commonly diagnosed very late and most doctors (cardiologists as well as neurologists alike) do not have a lot of experience with such patients. So I believe that he has told you that a neurologist should treat you as he might not have a lot of experience and not feel self assured enough. Also a neurological consult is recommended at least once to screen for signs which may indicate of a neurological condition of which autonomic dysfunction may be a part of such as multiple system atrophy (unlikely at your age).
So ideally because as I told you not all neurologists have experience in this field it would be best to be followed by a physician with a subspecialty in autonomic disorders (could be a cardiologist or a neurologist). That may not be that easy to do, depends on whether you live close to a big university medical center which usually have doctors with such subspecialties. If that is not the case, if you do not have access to such centers, I would wait first to see how it goes with the measures prescribed by your cardiologist (socks, salt etc) since the initial step would be that anyway. If you see no improvement I would then see a general neurologist.

I remain at your disposal for other questions.

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