Was the division of the Holy Bible into the Old and New Testaments divinely inspired?

There is no statement in the Bible anywhere to the effect that God calls for a literary division / distinction to be made between what is commonly called the Old Testament, and the New Testament. Given that no printing presses had been invented, there only ever were collections of various manuscripts (MSS) which were added to over the centuries. Clearly, until it became obvious to all that an end, or completion, had been made of inspired writings, no written statements could be made about any division.

However, there were natural divisions that existed over the centuries, for all to see. The Hebrew MSS were carefully kept and copied by Hebrew scribes. They began with the writings of Moses, they continued with historical documents about the nation of Israel, then psalms were written, then prophetic writings came along later. However, there was a 400 year gap between the last of 'the Prophets' and the birth of Jesus Christ. Yes, the Israelites continued to have writings about their history during that 400 year span, plus other MSS purporting to have spiritual content and miracles. But the Jewish Council of Jamnia never had those later MSS included in their canon of inspired writings. As far as the Jews were concerned, the sacred Hebrew scriptures began with Genesis and ended with Malachi.

Of note are the closing words of Malachi the prophet:

"Lo, I am sending to you Elijah the prophet, before the coming of the day of Jehovah, the great and the fearful. And he hath turned back the hearts of fathers to sons, and the heart of sons to their fathers, before I come and have utterly smitten the land." (YLT)

Now consider how the start of the Christian Greek scriptures pick up from where Malachi left off, by explaining the role of John the Baptist, the forerunner of the Christ, to prepare the Jewish people to receive their Messiah. Jesus was clear about this too.

Even before the last of the Hebrew writings of sacred scriptures, the Hellenisation of the known world had caused the Greek language to come into vogue and increasing use. That is why, when Jesus was born, Greek was already the language of Empire; legal and commercial documents all being in the Greek of the day. That is why, after Christ's death and resurrection, the MSS that now comprise the Christian Greek Scriptures were all written in Greek.

Finally, the word 'Testament' has long been understood with English legal documents that are headed, "The Last Will and Testament of...." to be the written instructions of a person as to how those who survive him are to be dealt with (materially). Because the death of the Testator effects the carrying out of his stated will, his "Testament" actually refers to a covenant he had made before his death. This is significant for the naming of the two sections of the Christian Bible.

The Hebrew scriptures deal with God and his covenant people. God states his will and the requirements of those who agree to enter into his covenant. They testify to their agreement; God spells out rewards for keeping his requirements, and curses for disobeying them. Further, the Hebrew scriptures speak of a time when God will make a new covenant that will not be written on tablets of stone. The Gentiles will be included in this new covenant.

The Greek scriptures deal with this new covenant, inaugurated by the death of the Messiah, and give the account of the continuation of God's dealings with his people and the Gentiles, after that 400-year gap.

This all adds up to reasons why, when printing of books happened, a statement would be inserted at the end of Malachi saying "The End of the Old Testament" (or similar) and at the end of Revelation it would read, "The End of the New Testament". Some translations have "The New Testament" just before Matthew's gospel account starts.

So, although it's an understandable and reasonable notation to distinguish the Hebrew from the Greek MSS, and the phrases 'old covenant' and 'new covenant' are in some of those MSS, it could not be said to be a 'divinely inspired' division, as you ask. It is a sensible, man-made notation. I don't see any link between that and the text you quote, John 13:34-35, however. I would have thought the texts in Jeremiah 31:31-33 and Hebrews 9:15-27 appropriately mention those two 'Testaments' or 'Covenants'.

Christians know that all of the Old Testament is vital to understanding all of the New Testament, and that all the apostles and Jesus himself quoted from those ancient Hebrew scriptures, to support their testimony that the promised Messiah had, indeed, come to fulfil all that was written about him. The old flows seamlessly on into the new, despite that 400-year gap.