Who first distinguished number theory and numerology?
Looking for methods is theory, looking for results is... whatever. The interest in pythagorean triples or magic squares seems to have evolved from knowing some items to knowing a method to generate all of them. But that does not preclude a motivation e.g. some kind of power, magical or other, to be derived: knowledge is power.
A similar question aks when astronomy separated from astrology and there is no good answer. According to Westman Copernic was still looking for a better astrology, but in his approach he was following Ptolemy who had already separated the method, that is astronomy, from the application - astrology.
It seems obvious that Euclid's proof about the infinity of primes is number theory, so at least this a terminus ante quem. Ever since Burkert's book the view about so called pythagoreans has radically changed and currently Zhmud has worked out details various writings. Also one should remember that little is known about ancient Greek isopsephy which became popular at the same time when neopythagoreanism was in vogue.