When did the last formal and recorded duel take place?

Does this count?

http://www.youtube.com/watch?v=zyzt2pONKaY

It happened in Kiev, 2013.

Or this:

http://www.yapfiles.ru/static/play.swf?st=vMDA2NDY1NTAt1335

In happened in St.Petersburg, 2013.

I will use this definition of a duel because it goes into the cause for dueling:

Dueling for honor is not the same as feuding, vendettas, brawls, jousts, or tournaments. It is “a fight between two or several individuals (but always equal numbers on either side), equally armed, for the purpose of proving either the truth of a disputed question or the valor, courage and honor of each combatant. The encounter must be decided or accepted jointly by both parties and must respect certain formal rules, be they tacit, oral or written, which will give it the weight of a legal proceeding, at least in the eyes of the two adversaries” (Billacois, 1990, 5)

The quote is taken from this paper that I have only skimmed for info on the Paris incident.

A few examples spring to mind:

The questions mentions a duel in the 60ties in France, Eugene Seidel helpfully provides a translation:

On 20 April 1967 during a heated debate in parliament, socialist deputy Gaston Defferre accosted Gaullist deputy Ribière, telling him to "shut up dumbass!" Soon after, Ribière sent him his seconds, demanding "redress by the blade". The duel took place at a house in Neuilly-sur-Seine and was refereed by another deputy, Jean de Lipkowski. Defferre insisted on real weapons and did not agree to end the duel on first blood drawn. The inexperienced Ribière was wounded twice. Finally Defferre consented to the referee declaring the duel over.

In the Book "The Secrets of Cabales Serrada Escrima" by Mark V. Wiley, about a Filipino martial art, students of one master Angel Cabales who taught them in California in the seventies through eighties recall that he told of duels with sticks that he, Master Cabales, fought to death in the period shortly after WWII. Maybe asking the same question on Martial Arts SE would yield some informed answers. The incident, while Cabales worked as a sailor, is described as follows:

Aboard the ship, Cabales became involved in an altercation that led to the coining of his slogan: "Three strikes and a man will fall." One day he was approached by a man claiming to be an escrimador and was asked if he would like to "practice." He allready knew what to expect because "practice" as in "try out", in those days meant a fight to the finish. Without hesitation, Cabales obliged the man, and with the third motion of his stick the man fell, never to get up

The story is told as a duel, but on should doubt this account - the event seems fairly spontaneous, the rules are liberal enough to be non-existent, and while Cabales may or may not have seen this incident as a duel, it probably is none in light of the definition given above. Note that both Cabales and his student (who wrote the book and thought highly of his teacher) have an interest in putting the incident in a somewhat positive light.

The last one is not documented but hearsay: A former boss of mine was in a student fraternity in the late nineties to early naughties. In Germany fraternities, esp. Burschenschaften, draw big on traditions from the 19th century (and are fairly conservative to right wing). Part of the tradition of some fraternities is a sort of dueling: Bouts with sabers, most of the body, the neck and the eyes are protected but it is expected that the face is injured and that scars may remain. These bouts are generally not about winning or losing or settling scores, but an opportunity for both participants to "prove their manliness."
Now, this former boss of mine once "insulted" someone from another fraternity by asking the guy wether he was a Jew, the other guy responded by challenging my boss to be to a duel - by having a junior of his fraternity delivering a ripped calling card to the fraternity of my boss. I would say this qualifies as a duel as opposed to a weird ritual since it was about settling a score.
Note that German fraternities always stress that these bouts are not about settling of scores, because then they would be criminal - So I assume no public records exist. I would still argue that they are duels, despite the lack of wide publicity, because of the strict rules, the (selected) audience of other Burschenschafters and the aspect of fighting for honor. I do believe that incidences like my boss told me are an exception, and most bouts are purely a rite of initiation into the fraternity.