A puzzle about European bank debt

There is something about this story about bank debt buybacks that I don't quite understand, although I have only had two cups of coffee as of the time of writing:

"European banks are turning to buying back their own debt in order to raise some of the billions in extra capital required by regulators. At least six major banks have launched debt buybacks in the last two weeks and investment bankers say more are likely."

Okay, so if a bank has debt - ie, others are lending it money - and the bank buys back, or in other words, pays off some of that debt, like paying off a credit card, say, how is this raising capital? The bank is presumably paying the debt off with, er, what? Fairy dust?

"In Lloyds' case, it will exchange bonds previously issued for new instruments that are compatible with new regulations. The move allows lenders to book profits and reduce the stock of non Basel III capital on their books without issuing new equity or offloading assets."

This is not very clear. What is the defining characteristic of "Basel III capital" in this case?

Finally we get a glimmer of how this actually works:

"The capital raised in this way is likely to be in the hundreds of millions. It boosts earnings by realising "own credit" gains that are otherwise purely theoretical. The market price of banks' debt has fallen dramatically in recent weeks, which enables banks to buy back their debt for an amount above the market price but below the cash they raised by selling the instruments, booking a profit."

Now I understand - I think.

As usual, the CityAM publication has a blisteringly good item on the Eurozone's latest absurdities today. It is become my daily morning read. The fact that several of its writers are friends and acquaintances is, of course, purely coincidental.