What is the pre-schism Christian church called today?

This is the book you want to read. Although it was in decline by the 11th century, the largest church at one time was the Church of the East. I don't remember if it was based on Arianism, but it was started by a heretical bishop who was kicked out of the Eastern Roman Empire and spread all the way to the Pacific.

At the end of the book the author, who's a real historian presents some evidence that it served as the inspiration for Islam as well.

The Lost History of Christianity

And reading one of his other books, there was never one Church. The original Church was the Egyptian Coptic Church. It may have been one official governing body that ruled all the churches but there were many differences between different churches in early Christianity that often ended in violence between the different sects.

That is in this book

The Jesus Wars

What is the pre-Schism Christian church called today ?

A far better question would be :

Why were the pre-Schism Christian churches able to coexist, despite their mutual theological differences¹, as a unified whole ?

to which the best possible answer is that they simply weren't aware of them, since church history shows that, whenever they took note of the existence of such obstacles, long-standing schisms would eventually ensue :

• Thus, at the third ecumenical synod, a significant part of the Middle East broke off from its other historical and apostolic peers, and the same happened two decades later, during the fourth ecumenical council, with most of North Africa and the rest of the Middle East breaking off from their remaining European peers, so that less than half a millennium after its inception, Christianity was already split into three main faiths.

• As for the Europeans themselves, they too would break off from one another about half a millennium later, splitting the by-then-mostly-Christian continent into its Eastern and Western halves, the last one of which would also split, yet another half a millennium later, along its Southern-Romance and Northern-Germanic ethnic lines. (Please note that traditional Protestant churches, such as, say, Lutherans or Anglicans, still bear a strong resemblance to their other apostolic European peers).

¹ _{These differences, albeit tacit, were already there. Africa and the Middle East have always held slightly different approaches with regards to Christology, just as East and West have always held mildly different approaches concerning ecclesiology and the procession of the Holy Spirit.}

Ignatius of Antioch
The earliest recorded evidence of the use of the term "Catholic Church" is the Letter to the Smyrnaeans that Ignatius of Antioch wrote in about 107 to Christians in Smyrna. Exhorting Christians to remain closely united with their bishop, he wrote: "Wherever the bishop shall appear, there let the multitude [of the people] also be; even as, wherever Jesus Christ is, there is the Catholic Church"

This designated the "true" or "universal" (community) church from several sects, schism & heresies

Question: What is the pre-schism Christian church called today?

I would argue that the first church council at Nicea produced the Nicean Creed and it coins the term, "holy catholic and apostolic Church".

"We believe in one holy catholic and apostolic church".

Eastern Orthodox and Western Roman Catholics have restated this creed as a profession of beliefs since the fourth century AD. Thus the pre schism church is properly called the catholic church with a date or text next to it which denotes it as pre-schism.

Less anyone thinks this answer expresses a Roman Catholic bias, I would argue it was the Western Roman Catholic church which came under greater influence of the pagan invaders which more diverged from pre schism Catholic doctrine. The Orthodox Church remains even today (or last time I looked at it closely) closer to the original unified Catholic church, pre schism. Given that, it's not so much that one church left the other, but that distance, foreign influences, and politics resulted in both east and west separating from each other along the lines controlled by the Byzantine Empire.

First of all, @LangLangC's answer is excellent. I intend only to expand on it.

The unified Eastern (Greek-speaking) and Western (Latin-speak) churches called themselves "orthodox" as opposed to the many heresies (like Arianism). They also called themselves "catholic" -- universal. They were united in religion, but administratively united only in theory, as they were much more fragmented administratively than we, today, think a unified church would be. Travel was hard even in the still-thriving Eastern Empire, and in the West, travel was very slow.

I expect that the old language of being "in communion" probably better describes the situation. Each city had its bishop and one or more churches. So you'd have the church of Milan, the church of Hippo, the church of Antioch, etc. The churches which considered themselves orthodox were "in communion", that is they recognized each others' sacraments, shared scriptures and confessions, recognized each others ordinations and accepted each others' members when travelling. They generally adhered to one or another of the ancient patriarchates.

So the church then was a much more decentralized thing than the church today. Note that this was mainly the effect of difficult travel -- especially in the Byzantine Empire the Emperor and the Patriarch of Constantinople exercised as much control as politics would let them. In the West, Rome's authority was necessarily entirely moral, and the East tended to forget that Rome still existed.

Some of the split between East and West is nearly as ancient as the Church itself. There was always a split in language and Constantine, the same Emperor who legalized the church, also was the Emperor who split the Empire!

Now fast forward six hundred years as the East climbs to the peak of Byzantium, while the West sinks to the worst part of the Dark Ages -- and then it all reverses, with Islam conquering one piece after another of the Eastern Empire, and the new, vibrant Medieval civilization is growing in Western Europe. The East resents the barbarian West trying to tell it what to do. The West resents the effete, snobbish East trying to conquer it and tell it what to do. And communication between the two got even worse as the Mediterranean got even more infested with pirates and travel grew even more difficult.

But everyone agreed that they were one church -- as long as you don't try to tell us what do, but do follow our instructions…

The final schism in 1054 was simply the last step of administrative separation when the hierarchies of the Eastern and Western churches excommunicated each other. (They were no longer "in communion".) This was basically an administrative thing. They continued to recognize each others' sacraments, ordinations, etc., but declared it to be unlawful to participate in each others' services.

As it happened, the Eastern church kept the title "orthodox" and the Western church kept the title "catholic" -- as far as I know, that was essentially historical accident, as the modern Roman Catholic and Eastern Orthodox churches both describe themselves as "one, holy, catholic and apostolic" and as "orthodox." (It's possible that the East called itself Orthodox because the richer East was the source of most of the heresies and orthodoxy was more of an issue there.)