'analytical or discriminating doctrine' is an early name for the original Buddha doctrine, called Theravāda. -
The term vibhajja-vādī occurs in M.99 and A.X.94, though not in the sense of a separate school, but as a characteristic of the Buddha himself: "Now, by blaming what is blamable and praising what is praiseworthy, the Blessed One is a 'discriminating teacher' (vibhajja-vadī) and is not one-sided in his teaching" (A.X.94).
Buddhaghosa, in the introduction to his Com. on the Kathāvatthu, says that in Asoka's time, when the Sangha prospered, many heretics took ordination as Buddhist monks but continued to spread their wrong doctrines. For purifying the Sangha, Asoka, together with the venerable Moggaliputtatissa, summoned assembly of the bhikkhus. When each of the assembled was individually questioned by the king about what the Buddha taught, those who said that he was an eternalist (sassata-vadī), etc. were expelled. The genuine bhikkhus replied that the Buddha was a vibhajja-vadī, an 'analyst' or 'discriminating teacher'; and when, on the king's question, Moggaliputtatissa confirmed that this was the correct view, those monks were admitted to the Uposatha (q.v.) assembly of the Sangha, and from their midst the participants of the 3rd Council at Pataliputta were selected. -
See Mahāvamsa, tr. by Wilh. Geiger, Ch. V, v. 268f.