The earliest "large scale" schism in the Christian Church was not the Schism of 1054, but at the Council of Chalcedon in 451 when the bishop of Alexandria and a number of other bishops were cut off for their adherence to a doctrine known as monophysitism. To this day, that separation remains and they are known as the Oriental Orthodox Church, predominantly spread across the Middle East, Africa, and parts of Southern India. They maintain that their belief is not monophysitism, as condemned by the Council of Chalcedon, but something called miaphysitism.

Even prior to that Schism, there was a smaller schism at the Council of Ephesus in 431 when the Nestorian heresy was condemned. This created what is known today as the Assyrian Church of the East (also, the Chaldean Catholic Church, which is loyal to Rome, was part of this continuity).

So, like today, what early Christians called themselves would have been contingent on the present company. Even today, in almost entirely Catholic nations, Roman Catholics will usually refer to themselves as Christians, not Catholics. In the southeastern US, where evangelical fundamentalism is most common, saying you are "Christian" usually means you are an evangelical fundamentalist. Likewise, starting in the 400's (or even earlier during the Arian heresy period), Christians would have been required to label themselves based on their loyalties to specific doctrines. For instance, "I am Chalcedonian" or "I am of Nestorius" would have likely been used when identifying allegiances.

My point is that Christians have, for the vast majority of their existence, been required to label themselves in order to identify what they believe, but they likely would not do that when speaking in a context consisting entirely of like-minded believers. At home and in their churches, they would be and are just "Christians." In public discourse and debate, they are more specific.

Also, the word "Catholic" simply means "Universal." Several churches use that word in their name, including mine. I am Eastern Orthodox (the other side of that 1054 schism), but our church's official name is "The Orthodox Catholic Church." With that in mind, it would technically be more accurate to shorten "Roman Catholic" to "Roman" rather than "Catholic."

'Catholic' has been an adjective describing the Roman Church since the earliest fathers, that is writing from early 2nd century CE. As mentioned elsewhere, 'Catholic' meant roughly the 'universal' Church.

However, the term 'Catholic' as a noun in and of itself, meaning 'I am a Catholic' or 'We are Catholics' seems to be a product of the late 12th/early 13th century, at the cresting of waves of orthodox renewal within the western Roman Church. In fact, the Canons of Lateran IV c 1215, which established so much of what we now think of as Catholicism, uses the noun 'Catholic', in particular in Canon 3 which reflects on the punishments to be meted out to heretics. For example:

Catholics who have girded themselves with the cross for the extermination of the heretics, shall enjoy the indulgences and privileges granted to those who go in defense of the Holy Land.

This was a new development.

It would seem then that the self-identification as 'Catholic', as metonymy of the institution, came at a time when the Roman Church was at the height of both its power politically and as orthodoxy was defined: order in the clergy was to be restored, spiritual power vs. temporal power settled, and the Roman Church's belief system concretized (for example, the sacraments finalized).

To be Catholic took on specific meaning in the 13th century, and it was enforced by clergy, monarch, inquisition. It was rare to find someone referring to themselves as a Catholic before this time.

I don't have a comprehensive answer for you, since early church history doesn't fall within my immediate purview of history, but I can say that the word "catholic" is an adjective essentially meaning "universal or general" and it was employed in the Creed of First Council of Nicea (325). The phrase "Εἰς μίαν, ἁγίαν, καθολικὴν καὶ ἀποστολικὴν ἐκκλησίαν·" (In one, holy, catholic and apostolic church) was also used in the Creed of the First Council of Constantinople (381).

Based off of this evidence, we know for certain that ecclesiastical leaders consciously decided to describe their religion as "catholic" - or all encompassing. I suspect that it had been used earlier than these two instances, but I am hesitant to say anything authoritatively.