The New Covenant Foretold
The key to the division can be seen in Jeremiah, where God speaks of the old covenant and the new covenant. Israel had broken the old covenant, so God spoke of a new covenant.
31 “Behold, days are coming,” declares the Lord, “when I will make a new covenant with the house of Israel and with the house of Judah, 32 not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them,” declares the Lord.
33 “But this is the covenant which I will make with the house of Israel after those days,” declares the Lord, “I will put My law within them and on their heart I will write it; and I will be their God, and they shall be My people. 34 They will not teach again, each man his neighbor and each man his brother, saying, ‘Know the Lord,’ for they will all know Me, from the least of them to the greatest of them,” declares the Lord, “for I will forgive their iniquity, and their sin I will remember no more.” Jeremiah 31:31-35 NASB
The Breaking of the Old Covenant
The enumeration of covenants includes those with Noah, Abraham, Moses and David, among others. So, if we were being precise, we might say the "Old Testaments" (plural) and the "New Testament (singular). However, the primary covenant that governed all of Israel was probably the Mosaic covenant, which the people broke. The book of Hebrews affirms this:
6 But now He [Jesus] has obtained a more excellent ministry, by as much as He is also the mediator of a better covenant, which has been enacted on better promises. 7 For if that first covenant had been faultless, there would have been no occasion sought for a second. 8 For finding fault with them, He says,
“Behold, days are coming, says the Lord,
When I will effect a new covenant
With the house of Israel and with the house of Judah;
9 Not like the covenant which I made with their fathers
On the day when I took them by the hand
To lead them out of the land of Egypt;
For they did not continue in My covenant,
And I did not care for them, says the Lord.
10 “For this is the covenant that I will make with the house of Israel
After those days, says the Lord:
I will put My laws into their minds,
And I will write them on their hearts.
And I will be their God,
And they shall be My people.
11 “And they shall not teach everyone his fellow citizen,
And everyone his brother, saying, ‘Know the Lord,’
For all will know Me,
From the least to the greatest of them.
12 “For I will be merciful to their iniquities,
And I will remember their sins no more.”
13 When He said, “A new covenant,” He has made the first obsolete. But whatever is becoming obsolete and growing old is ready to disappear.
Jesus, the Mediator of the New Covenant
Jesus Christ is, thus, the mediator of the New Covenant, as Hebrews 12 affirms:
But you have come to Mount Zion and to the city of the living God, the heavenly Jerusalem, and to myriads of angels, 23 to the general assembly and church of the firstborn who are enrolled in heaven, and to God, the Judge of all, and to the spirits of the righteous made perfect, 24 and to Jesus, the mediator of a new covenant, and to the sprinkled blood, which speaks better than the blood of Abel. Hebrews 12:22-24 NASB
The Inception of the New Covenant
This division is affirmed in several places. It was introduced by Jesus at the Last Supper.
23 For I received from the Lord that which I also delivered to you, that the Lord Jesus in the night in which He was betrayed took bread; 24 and when He had given thanks, He broke it and said, “This is My body, which is for you; do this in remembrance of Me.” 25 In the same way He took the cup also after supper, saying, “This cup is the new covenant in My blood; do this, as often as you drink it, in remembrance of Me.” 26 For as often as you eat this bread and drink the cup, you proclaim the Lord’s death until He comes. 1 Corinthians 11:23-26
So, everything does change with the coming of Jesus. The New Testament is the collection of all the books that being with the life of Jesus, the Mediator of the New Covenant.