What's the smallest conflict to date that we refer to as a "war"?

If you're asking about wars with the fewest casualties, there are plenty of conflicts in which not a single casualty was recorded (search for "bloodless wars"). Most of these situations were caused by formal declarations of war or nations entering into states of war without actual hostilities from either side.

One of these is the Three Hundred and Thirty Five Years' War, between the Isles of Scilly and the Netherlands. The Isles of Scilly are located southwest of Cornwall, and were a base of the Royalist fleet during the English Civil War. The Dutch navy (allied with the Parliamentarians) declared war on them, never attacked, and then neglected to make peace. (Whether this was a formal declaration of war, or if such a declaration occurred at all, is a matter of debate, as it was allegedly declared by an admiral, not by the Dutch state. But the Dutch ambassador in London did travel to Scilly to sign a peace treaty.)

Another such a war was the Anglo-Swedish War between 1810-1812. Having defeated Sweden and concerned that the country continued to trade with Britain despite the Continental System, Napoleon forced Sweden to declare war on Great Britain. Since neither Sweden nor Britain had any actual intention of attacking the other, no war was actually fought. (In a tragic turn of events, Sweden did fear a possible British attack, and ten farmers were killed in a protest against the resultant conscription.)

The shortest "war" on record that I know of is the Anglo-Zanzibar War, which was fought on 27 August 1896 from about 9:02 to 9:40. In other words, the war was about 38 minutes long.

There have also been plenty of local conflicts that have been called wars, such as the Lincoln County War.

If you're looking for short AND bloodless, look no farther than the Honey War. "Fought" from about August-December 1839, it was a hot-blooded border dispute between Missouri and what is now Iowa. The situation came to a head when some trees were cut down in the disputed territory to harvest their honeycombs. Among the more amusing details is this description of the Iowa militia who arrived to defend "their" land:

There was no shortage of officers, as the group included four generals, nine general staff officers, forty field officers, and eighty-three company officers. The troop provided their own weapons which included blunderbuses, flint locks, a plow coulter connected to a log chain, a six foot sword cut from a piece of metal, pitchforks, many swords from the War of 1812, and even a dasher from a churn.

Both camps of militia suffered from low supplies but high spirits (in some cases literally, as they were given free whiskey by their officers). In one incident, a Missouri sheriff was arrested and put in jail in Iowa; Missourians believed him to have been kidnapped and wanted to free him, but fortunately nothing came of it. In the end, leaders on both sides had to struggle to keep the rowdy militias from going at each other. As Thomas L. Anderson of Missouri, trying to set up a commission to sort things out, said, "And in the name of God of mercy and justice gentlemen, let this monumental piece of absurdity, this phenomenal but cruel blundering have an end."

In the end, cooler heads prevailed with a join resolution between both governors on 12 December 1839. The Missouri militia got word that peace had been made without even a fight. To express their anger that they hadn't had a chance for a good scrap, they shot a deer and labeled each half with the name of a governor. These halves were then "riddled with bullets" and buried with mock military honors, and the militia left in a huff. The Iowans, not receiving the news, sent a scouting party to see what was taking the Missourians so long to attack. When they found no enemy there, they were told of the peace, and they, too, departed in disappointment. Tensions continued to flare until a final resolution was made about the border in 1849. A line of pillars laid down to mark the boundary was completed in 1851.

Major source: Hill, Craig. "The Honey War" Pioneer America. 14.2 (July 1982): 81-88.

PS I should point out that what is referred to as a "war" is arbitrary. Clearly, the Honey War was a war in only name, though "we" do refer to it as such, because this was the name given to it at the time, partly as a way of poking fun at it in the newspapers. Likewise, The Three Hundred and Thirty Five Years' War was only named long after any actual hostilities could have taken place. "Peace" was made in 1986.