What an excellent question! And a very complex one that has been researched, and continues to be researched, by Professor Dauvit Broun at the University of Glasgow.

Let's begin. The kings of Scots became earls of Northumbria by marriage: King David I married Matilda of Senlis (or Matilda, daughter of Earl Waltheof of Northumbria) in 1112. Matilda was the heiress to Northumbria and also was the countess of Huntingdon, an English honour in Cambridgeshire. King David then became earl of Northumbria by marriage. So, the kings of Scots were kings by hereditary right but they were also English earls initially by marriage and later by inheritance. Professor Broun has researched the self-identification of the monks of Melrose Abbey during the end of the twelfth century and the beginnings of the thirteenth, and evidence from the nearby Dryburgh Abbey, particularly from a chronicle written by Adam of Dryburgh, who wrote in the last decades of the 12th century, suggests that Scottish monks considered themselves to be living in the land of the English but the kingdom of the Scots. It is worth suggesting, then, that Dryburgh's remark encompasses the self-identification of other people living in this liminal space.
In 1173, the king of Scots, William I The Lion (David's grandson), becomes entangled in what is known as the Young King's Rebellion: Henry II of England's enemies, like the king of Scots and the king of France, joined Henry's son against the king. King Henry had taken Northumbria and the honour of Huntingdon from William and he wanted to recover his inheritance. Anyway, things didn't go well for William and he was defeated and taken to Henry (his legs tied to a horse, no less) where he was forced to pay homage to Henry as his overlord. This is called the Treaty of Falaise of 1174. This "overlordship" had no precedent in history--in fact, Henry didn't claim his overlordship had a precedent as far as I'm aware--but it seems to me that it is possible that the drastic change in Anglo-Scottish relationships at this point did influence the way peoples in the Scottish Borders felt about their identity. We know that, by the mid-twelfth century, people in the Scottish Borders considered themselves Scots. One of the things I argued in my PhD thesis was that the Chronicle of Melrose Abbey's portrayal of King Malcolm III in 1173-4 was conditioned by King William's new status as a client king, which in part shows the monks' self-identification as English. But my the late thirteenth century, a scribe from Melrose made several changes to the manuscript that "corrected" the notion that Malcolm was a client king of England, stressing how he was king of Scots by hereditary right, and therefore, so were his descendants, Alexander II and III. So, to summarize all of this, the answer is that there is a change in self-identification in the Scottish Borders sometime between the last quarter of the twelfth century and the mid-thirteenth century. But when exactly did this happen? We are not entirely sure. How gradual was the process? Good question.