When did they take on a new name to distance themselves from heretical churches?
First of all this “reformation” that you relate to deserves some defining and clearing up.
In the West the reformation dealt with disciplines and abuses, such as “abuse” by Luther’s own German priest abusing indulgences. The Church reformed these abuses, and reformed the disciplines of the priest hood to live and follow the gospel, as St.Francis of Assisi, St.Ignatius of loyola etc…
What resulted from the protestant reformation came bodily divisions such as Lutheranism, Anglican Church, Anabaptist etc…
Early on in the Eastern Church went through radical reforms in doctrine and disciplines. Disciplines in the form of “Monasticism”, when the Eastern Church Fathers refused to incorporate the Roman Emperor rule on Church matters, much so that these Fathers left their positions into monasticism.
Then the East encounters reform from “discipline of thought” stemming from the Alexandrian way of thinking from the Antiochene way of thinking, although both agreed on Jesus divinity and humanity, their differences was in understanding this union.
The Eastern Church reformed in matters of doctrine when she came under attack by her own bishops such as Arian, Nestorius and the Eastern Monk Eutyches.
From these came their excommunications and heretical teachings of Arianism, Nestorianism, and Docetism. From these the East takes on more heterodoxical teachings from “Monophysites” branched into the Jacobites which conflicted with the Chalcedonian definition of Christ nature. This is huge as far as reformation is concerned in regards to doctrine from the Eastern Church’s.
The Western Church did not have these “doctrinal” difficulties in regards to Christ’s nature. Tertullian in the West was able to almost single handedly bring the great intellectual tradition of the East to the West, who did not have a problem with Jesus divinity and humanity.
What hardly gets mentioned from this Eastern Church history is that from their Arian heresy continued and infected the Arabs (the founder of Islam) to recieve a heretical view of Christ as not being God which continues today.
Orthodox was a title taken from the Early Church councils who ruled between what was Orthodox “right teaching” from the Eastern counter parts who refused to obedience to the Councils findings who remained and taught Heterodox “wrong teaching”.
It should be noted here that the title “Orthodox” was never put in place to separate the Eastern Church’s from the Popes. This title Orthodox used today has taken on a new “unorthodox” shaded meaning to divide the Orthodox from the Popes. The problem is that many of their sister heterodox Church’s have also taken on the Orthodox title today.
From the council of Ephesus 431 a.d St.Cyrils second letter to the heretic Nestorius was declared to be harmony with the Council of Nicea and thus “Orthodox” made its way into the Eastern Church’s who held to the Nicea/Constantinople councils hold to the Orthodox teaching of the Church, while those Church’s in the East who refused the early Church councils findings such as Nicea were distinguished from one another by Heterodox.
I believe in one of these councils a Popes letter of decree also announces what is held in the title “Orthodox”. Sorry can’t pin point it off the top of my head at this moment. But the title Orthodox begins to be taken on from these early councils to identify themselves from the Eastern heterodox Church’s who mirriored them in liturgy, language, culture.
What is interesting to note here is that “indulgences” are still practiced in the Catholic Church what was reformed was the abuse of indulgences. From these matters resulted in protestants leaving the Catholic Church.
In the East, Jesus nature came undertack because other Eastern Church’s were teaching multiple different natures of Jesus, which also caused division among the Eastern Church’s ranks. In the West we call them who left the Church Protestants or separated brethren, in the East originally disciphered one another (not separated from the popes) as Orthodox and heterodox.
Peace be with you