The term Turcopole derives from the Byzantine Greek Τουρκόπουλοι, translating to "sons of Turks", as you noted. Originally, the term was used to designate any group of soldiers of Turkish descent in service of the Byzantine Emperor, but as the overwhelming power of the Empire declined following the death of Emperor Manuel I Komnenos in AD 1180, these groups of Turkish warriors became more and more groups of scattered mercenaries for hire in regions in and around Asia Minor. They were often armed in the Byzantine style and well-trained for a cavalry role, and mostly came from Eastern Christian backgrounds. They saw extensive use by the Crusader states, not just the Templars, and were renowned for being capable warriors.

The position of turcopolier was probably similar to the way the rank of akolouthos or the komes tes hetaireias worked in the Byzantine military heirarchy, since many of the Holy Orders had origins during the First and Second Crusades, where the Byzantines played a major role in establishing and protecting the Outremer. The aforementioned positions were relatively high-ranking (soldiers generally of indigenous Byzantine descent) who were put in charge of managing the foreign corps of troops, the Varangian Guard, the Tourkopouloi, etc. In the same way, I would assume the turkopolier was a native Frank who was put in charge of managing the affairs of the Turcopoles and making sure the battalion remained loyal and battle-ready for whatever Holy Order they served.