It is conventional to number the three persons of Latin and Greek and many other languages so that the first person is the speaker, the second one is the listener, and the third one is anyone else. Did the ancient grammarians already have this numbering? If yes, was it an explicit numbering (first, second, third) or a more implicit ordering (always giving things in the order facio–facis–facit)?
The numbering is most importantly a convention and it makes life easier if everyone sticks to the same nomenclature. I don't know if the modern convention goes back to the antiquity. In the plural the numbering also describes an order of precedence in Latin and Greek: any group containing me is "we", any other group containing you (sg.) is "you (pl.)", and any other group is "they".
Both Latin and Greek sources are fine, as I believe the two languages to be essentially identical when it comes to persons. Any insight into the history of numbering of persons would be quite interesting.