I just finished a 23-hour drive (straight), so bear with me as I sort through these terms for you, which have meanings beyond their being titles.
It's important to understand that the words "catholic" and "orthodox" have general meanings as words outside a religious connotation which eventually came to be associated with religion, particular meanings when applied to the Christian faith, and now an additional special meaning as titles of the Churches they're called. You are correct in that both (what are known as) the Catholic and Orthodox Churches both claim to be at once catholic and orthodox in their particular senses, while being called apart from one another, respectively, Catholic and Orthodox. Generally, you'll know the sense they're being used in simply by whether they're capitalized - if they get the big "C" and "O" they're referring as titles to the Churches proper, and if they get the little "c" and "o" they're referring to the trait which both Churches claim.
Outside this use of the terms "orthodox" and "catholic" however, it's also important to understand that the word "Church" also has a particular meaning within Christian history and to both the Catholic and Orthodox Churches: whereas the word "church" can mean a building designated for religious worship and/or an assembly of the faithful, when applied within Catholicism and Orthodoxy in a particular sense it designates a body of the faithful who profess a common faith (e.g. the Nicene Creed) and who are united under a validly ordained and lawfully governing bishop. Thus the Catholic Church and Orthodox Church are more than simply a monolithic structure: they're vast organizations comprised of many smaller, autonomous-and-unique-but-nevertheless-united bodies spread throughout the world. The Catholic Church is not, for example, one blob headed by the bishop of Rome (the pope), but many independent particular Churches headed by their bishops who are all in communion with one another and at Rome. The majority of these belong to what's known as the Roman Rite, but there are also those which belong to the Greek (Byzantine) Rite, the Coptic Rite, the Maronite Rite, and so on. In the same way, the Orthodox Church is not one blob headed by the Patriarch of Constantinople, but many independent particular Churches headed by bishops, and so forth.
This plays into your question, because when you dig into the history of the terms themselves, they are used in their general sense before being applied to either Catholicism or Orthodoxy in their particular or special sense, and when they are used in this sense they refer to the Churches as constituted in that way.
So, what did they mean originally, and when did they start to be applied in a uniquely Christian sense, and then, eventually, as a special titular term describing both the Catholic and Orthodox Churches proper:
The word catholic derives from the Greek katholikos, which has a generally simple meaning of "universal" or "of the whole." While it doesn't originally find its origin in Christianity, it was used to definitively describe apostolic Christianity (which both the Catholic and Orthodox Churches stake claim to through lines of apostolic succession dating to the original apostles) in a particular sense by at the latest the very early second century, and in the special sense as a Church by the fourth century, as the Church described itself as such at the Council of Nicaea, periods during which when the Church was struggling with a number of errors and used the term to distinguish itself from groups which had embraced these errors. A primary example is Donatism and the Donastists, a group prominent in a part of Africa nominally associated with the Western Churches which arose out of the scandals caused by Christians who lapsed during periods of persecution and subsequently desired restoration of union once persecutions were over; the Donatists believed believed members of the clergy must be impeccable to exercise their ministry, and therefore any who had lapsed during persecutions must be excluded should they attempt to rejoin the Church - a view which was not shared by the wider (universal) Church, was condemned, and therefore Donatists could not consider or style themselves part of the Catholic (universal) Church.
The word orthodox is from the Greek orthos and doxa which combined has a general meaning of "right believer." It also doesn't find an origin in Christianity, but also came to be used to definitively describe apostolic Christianity in a particular sense by, at the latest, the fourth century, when it was used most correspondingly to the Arian heresy, distinguishing the correctness (orthodoxy) of the faith as again defined by the Council of Nicaea and later the Councils of Ephesus and Chalcedon against that held by Arius, Nestorius, and Eutyches and their followers, who were especially prominent in the East (and not orthodox).
Ultimately, it is likely due to this prominence (of the Arians et al) in the East that the term orthodox was more widely and vigorously applied to describe the Churches there in the following centuries, whereas the West leaned toward catholic as a result of its patrimony and struggles.
Of course, the Church which held the aforementioned Councils and which was described as both c/Catholic and o/Orthodox split in the 11th century, in what is now known as the Great Schism. At the time, however, it was not known as the Great Schism. It was one, in fact, of many schisms between Western and Eastern Churches, and at the time was not believed to be permanent (for a variety of reasons held by either side). Ultimately, attempts at reunion (notably the Councils of Florence and Lyon, still hotly debated) failed, and as the split became more ossified, both the Western and Eastern Churches developed their own terminology, largely independent of one another. During this time, the opposing Church may simply have been called the "Western Churches" or the "Eastern Churches," but the distinction between "Catholic" and "Orthodox" wasn't made, and you still have either side referring to itself as the "Catholic" and "Orthodox" Church...
...Which, in a technical sense, they still do, despite also using the terms Catholic and Orthodox to easily distinguish between themselves. All that said, there is no special decree by which either Church recognizes itself or the other as strictly Catholic or Orthodox. Rather, you have an evolving tradition within those communions to refer to themselves as such, e.g. the Catholic Church at the Second Vatican Council (mid-20th century) referring to itself as the Catholic Church in its official documents and the Orthodox Church at the Holy and Great Council of the Orthodox Church (at Crete, early 21st century) referring to itself as the Orthodox Church in its official documents...
...despite still acknowledging themselves as both Catholic and Orthodox in the particular, theologically significant uses of the terms.
So, yeah: both Churches consider themselves to be both catholic and orthodox in the particular sense, while recognizing the other as Catholic and Orthodox in a titular sense.
u/otiac1’s sleep-deprived explanation is better than anything I’ve been able to come up with fully awake, haha.
But if I can attempt to add to it, I would say that the situation is actually more complicated than simply “two churches”. After the schism in 1054 there were actually four churches, as two others had already split off centuries earlier. But all of them described themselves as “catholic” and “orthodox”, and there are other similar terms as well, such as “ecumenical”, which also means “universal”.
“Ecumenical” is also the term used to describe several late antique/early medieval church councils. At the time there were five major centres of Christianity in the Roman Empire: the old capital in Rome, the new capital in Constantinople, Alexandria in Egypt, and Antioch and Jerusalem in Syria. At that point the church really was lower-case catholic, orthodox, and ecumenical (more or less). There were disputes and sometimes schisms, such as the Arian schism in western Europe, but the Arians were eventually reunited with the rest of the church.
Those early councils are often all about what kind of being Christ was supposed to be. The earliest schism that turned out to be permanent occurred at the Council of Ephesus in 431. The council condemned “dyaphysitism”, the doctrine that Christ had two separate divine and human natures. Dyaphysites were also called “Nestorians”, since they followed the teachings of Nestorius of Constantinople. They’re also called the “Church of the East”, since dyaphysitism became popular among the Christians in Persia, who weren’t part of the Roman Empire. After 431 the Church of the East developed pretty much independently from the church in the Roman Empire.
The second schism occurred twenty years later in 451. Everyone else agreed hat Christ had one nature instead of two, but was that nature fully divine, or both fully human and fully divine at the same time? Some people argued that even through Christ became a physical human, he had only one divine nature (“monophysitism”) or maybe more accurately, only one divine nature that manifested itself through a human form (“miaphysitism”). In 451, the Council of Chalcedon condemned this and asserted that Christ was both human and divine at the same time, not separately, and this human-divine nature existed in one human body.
The churches in Constantinople and Rome agreed with this, so now we refer to them as “Chalcedonian” churches; the churches in Alexandria and Antioch did not, so they are considered “non-Chalcedonian”. The smaller local churches of Armenia, Georgia, and Ethiopia also eventually rejected the Chalcedonian position. Today we call these non-Chalcedonian churches “Oriental Orthodox”.
Meanwhile, although Rome and Constantinople were still united (more or less), they developed political and language differences, in addition to doctrinal disputes. Rome and Constantinople represented the two halves of the old Roman Empire, but in the 6th century, the western half no longer existed. The Roman emperor still ruled in Constantinople, so the church in Constantinople felt that the church had moved along with the emperor, and it was now the head of the entire church. Italy was ruled by Germanic kingdoms, but the Roman church felt that its authority had never been transferred to Constantinople, so Rome was still the head of the church. They also spoke Latin in Rome and Greek in Constantinople, and the Latin church used unleavened bread for the Eucharist while the Greek church used leavened bread.
The Latin church had also added the “Filioque clause” to the Nicene creed, the statement of faith that everyone had agreed on way back at the first ecumenical council in Nicaea in 325. The question of Christ’s nature was one thing, but they were also concerned with defining the Trinity (God, Christ, and the Holy Spirit) and how each component was related to the others. At Nicaea they decided that Christ and the Holy Spirit proceed from God, but by the 11th century the Latin church had added “and from the Son” (i.e, Christ), or “filioque” in Latin.
In 1054, ambassadors from Rome met with the Patriarch of Constantinople to discuss this, but they ended up insulting and excommunicating each other. It wasn’t actually a big deal at the time, and 40 years later the emperor asked the Pope for military aid, which ended up being the First Crusade. The crusades sort of emphasized the differences between the Latins and Greeks though, and another crusade ended up temporarily destroying the empire and the Greek church in 1204. But the schism wasn’t really complete until the Latins and Greeks tried and failed to reunite over several church councils in the 15th century.
So for your actual question, when did we start using the terms “Church of the East”, “Oriental Orthodox”, “Eastern Orthodox” and “Roman Catholic”, there isn’t really a specific date (as otiac1 noted) but it happened relatively recently.
For ancient and medieval Christians, they couldn't conceive of the idea that there were four different churches. There was just one church, and various heretical groups that could theoretically be reunited with the church. No one thought of the other groups as valid, separate churches. When they referred to each other with different names, they simply described them based on geography, or based on the name of their leader/founder. For example, I shouldn't say they're all recent terms, because “Church of the East” or the “Nestorian” church is actually the ancient terminology - named after where they were (far off in Persia, east of the Roman world), or their leader Nestorius. (The aforementioned Arians were also named after their leader, Bishop Arius.)
The church in Rome simply called itself “Holy Roman Church” - this is such a common formulation that it’s almost always abbreviated “SRE” (sancta Romana ecclesia in Latin). The Greeks called it the Latin church, and the Latins called them the Greek church. The Armenian church was for Armenians who spoke their own language, likewise for the Syrians and Georgians and Ethiopians. In Egypt the Christians called themselves “Copts” so they had a Coptic church. Syrians were also called Jacobites, after an early leader named Jacob, or Melkites, after the Arabic word for king (since they were subjects of the eastern Roman emperor). There were Maronites in Lebanon, also named after an early leader, Maro. Sometimes churches also called themselves “Apostolic” (Rome and Armenia for example), claiming a direct connection from Jesus’ apostles.
After Constantinople fell to the Ottomans in 1453, the Greek church split into national churches (Russian, Serbian, Bulgarian, etc), but they all followed the Greek rite of the church in Constantinople.
To add to the confusion, Muslims didn’t really know or care about any minor doctrinal disputes, so to them all Christian churches were “Roman”!
Still today, what we're thinking of as the “Eastern Orthodox” church officially calls itself “Orthodox Catholic”. But this is simply a convention, at least in English and other western European languages, to distinguish it from western Catholicism, which is sort of taken to be the default ancient/medieval church. This convention is "recent" in terms of the history of the church, but it started in the 16th/17th centuries, after the Protestant Reformation in Europe.
By then, first of all, it was clear that the schisms that occurred centuries earlier were never going to be healed. Secondly, the Protestant churches realized they had a lot in common with the eastern churches. They both felt they represented the ancient original form of Christianity, without all the abuses introduced by Rome. But then, the Roman church reformed itself as well, and some Orthodox churches ended up uniting with them. So today there are Roman Catholics, as well as Eastern Catholics, who are different from Eastern and Oriental Orthodox.
“Oriental Orthodox” is perhaps the only one of these terms that has a specific date. In 1965 they invented term “Oriental Orthodox” for a church conference held in Ethiopia. Even though this effectively means the same thing as “Eastern Orthodox”, it was their way of distinguishing themselves from the Greek-based Orthodox church.
Sources:
otiac1 has already noted the lack of specific sources, but I’ve found these useful:
Encyclopedia of Eastern Orthodox Christianity, ed. John A. McGuckin (Wiley-Blackwell, 2011), particularly Peter C. Bouteneff’s article on “Oriential Orthodox”
Kallistos Ware, The Orthodox Church, 2nd ed. (Penguin, 1993)