In the 7thC what is now the Scottish Borders and part of the Lothians itself up to the Forth came under Anglian control ruled from Bamburgh after they took over control from the previous rulers who were Britons. So this was called Bernicia which then joined with the more southerly Deira and formed the Kingdom of Northumbria. This included Edinburgh etc as well as the Borders.

Towards the end of the millenium things changed as most of what is now northern England became part of the Danelaw including that part of Northumbria which was previously Deira - that is from the Humber to the Tees approx. Bernicia or the northern part of Northumbria was nominally independent and controlled by the Earls of Bamburgh but Northumbrian power was much reduced with losing the southern part to the Danelaw at the same time as the emerging Kingdom of Scotland (Alba) was expanding south of the Forth. So it is thought that the Battle of Carham just reinforced what was already a reality that the northern part of Bernicia (the current Scottish Borders and Lothians) had come under control of the Kings of Alba. Bamburgh held on to the southern part of Bernicia (ie modern Northumberland and down to the Tees) until it was later absorbed into the emerging Kingdom of England.

I think the border could certainly have been further south - or even further north at that. And there were periods when Scotland controlled part of what is now England and vice versa - but it is generally regarded that (Berwick aside) the border in the east side of the country has been more or less as it is now since Carham.

The Scottish kings held land in England (as Earls both of Northumbria and Huntingdon) which they swore fealty for - which caused all sorts of problems later when the English kings tried to force the issue that the fealty was for Scotland itself.

It is an interesting area though. Here in Kelso we have the ruins of Roxburgh Castle which the Scots themselves destroyed so it could never fall into the hands of the English again. James II of Scotland was killed by his own exploding canon whilst retaking the castle and the young infant James III was crowned in Kelso Abbey!