Orwell's "You And The Atomic Bomb" essay, written in 1945, is predicated on the idea that it's technology that wins wars and technology is expensive and exclusive to large governments. Writing in October 1945 just after a war where technology played a very important role and just after the introduction of the atomic bomb, I don't blame him. To him we seemed poised on an era of new empires controlling the world with an untouchable army of heavy tanks, heavy bombers, and atomic bombs.
And that technologist view of warfare was almost immediately proven wrong.
A wave of colonial revolts employed asymmetric warfare to counter the technological superiority of their overlord's armies. The nukes were useless because nobody was willing to use them, and the technology designed for conventional warfare was countered with cheaper alternatives and guerilla tactics. France, Britain, and later the US and Soviet Union, were all defeated by (or had a very, very hard time against) citizen armies. 70 years later and we're still contending with this problem.
So I believe the whole premise has issues from a military history point-of-view.
By "fair weapon" I take to understand one that both sides have access to in adequate numbers to make it a "fair" fight. Not just one side roflstomping the other with air power or armor so thick nobody could hope to pierce it.
But there's more than just availability, there's training. Some weapons require more training to use effectively than others. You can train an effective unit of pikemen or musketeers, for example, relatively quickly. Longbows and swords, on the other hand, require years of training and physical fitness. They require that you have not just the money to buy the weapons, but also the leisure time to train.
This is a rather subjective measure, what is a "fair weapon" is specific to the situation.
Take muskets for example. Are they a "fair weapon"? If you're an American colonist in 1776, sure! You probably own at least one and have been using it your whole life. But what about an American native in 1776? Or native under the British Raj?
In medieval warfare the equivalent of the "totalitarian weapons" might be heavy cavalry, and heavy infantry: professional soldiers (or nobility with a lot of leisure time) with expensive equipment using formations that require a lot of training together. In modern warfare it's things like modern main battle tanks, modern combat aircraft, satellites, complex command and control systems... things that require large support structures and supply chains and a lot of money paid up front to get the best protection and the best gun.
In modern warfare the "fair weapons" would be cheap, effective counters to all that. The AK assault rifle; any number of man-portable anti-tank and anti-air weapons; the improvised explosive device (basically booby traps); cell phones with Internet and encryption. If you can use a $30 cell phone and some left over 155mm artillery shells to fashion a trap which damages a $9 million tank and wounds its highly trained crew, that's a fine trade off.
But even tanks and aircraft aren't exclusive to the "totalitarian" side. Arms from previous generations, particularly Soviet arms, are so ubiquitous and relatively cheap that a civilian army can get its hands on, for example, a few T-54s.
And then there's the notion that fights should be "fair". Fair fights are for suckers. In warfare you want the fight to be as unfair for the other side as possible. Hide your weaknesses. Trick them into attacking where you're strong. And attack their weaknesses with your strengths.
This could be using the political situation of your enemy to your advantage, perhaps exploiting unrest as in the Vietnam War, or taking advantage of complex rules of engagement. It could be economical, making it too costly for the enemy to win. It could be logistical, wait and starve them out. You can use your superior mobility and intelligence to avoid their military units and attack their supply lines. Your side might simply be willing to absorb more casualties than the other.
The whole notion is generally referred to as Asymmetric Warfare and it's what a successful civilian insurgency is likely to employ against a conventional army. A war is rarely won simply by the weapons of the two sides.
The comments brought up the issue of who "wins" a war. Determining who won or lost a war often covers up a complex situation and a lot of historical detail with squabbling over a label. War isn't a game. You don't get more points by blowing more stuff up and "win". Wars have goals, victory has to be worth those goals. And wars have aftermath that must be dealt with.
The Vietnam War is a good example of militarily "winning", yet losing the war. In terms of damage inflicted or casualties, the US "won" the war hands down, but what were their goals? I don't know that was even clear to the US at the time. If it was to defend South Vietnam they clearly lost, and why were they defending a dictatorship? If it was to "contain Communism" then they fought in vain. Domino Theory was incorrect. North Vietnam wasn't fighting to spread global Communism. North Vietnam fought to expel the foreign armies which had occupied Vietnam for generations and to reunite the Vietnam. The US could have gotten the same result by not getting involved in the first place.
Sometimes nobody wins, the War of 1812 is a good example. The US intended to stop the British from impressing sailors and annex Canada. The British wanted to set up a Native American buffer state in what was the American Northwest. The US was repelled from Canada. The British stopped impressing sailors, but only because the Napoleonic Wars were over; the US could have achieved that without a war. By the end of 1814 the war was getting too expensive for both sides to press home, so they negotiated a peace that put everything back the way it was.
Sometimes one side decisively wins the war only to find themselves saddled with a messy aftermath. In the Iraq War the forces of the Iraqi government were decisively defeated in just two months. Yet the US found itself in a messy insurgency for years and years to come.
That said, I can think of a few wars which were "lost" (or "won" at very great cost) by the side with the "totalitarian weapons", and won by the side with the "fair weapons".
Although this was fought by two professional national armies, there was a huge disparity in weaponry that I think makes it qualify. Russia had modern aircraft, artillery, and tanks in very large quantities, while Finland was mostly fighting with small arms and machine guns. Conventionally it should have been a walk over. While technically the Soviets "won" it took three months, and over 300,000 casualties. Finland remained free and the Soviets were humiliated.
This would be the Viet Minh fighting their French colonial masters. The Viet Minh fought with mostly small arms and foot soldiers (later artillery) against a modern Western army, navy, and air force. The Viet Minh won. They got the French to leave. They got their own country.
What started as the Democratic Republic of Afghanistan, requesting the assistance of the Soviet Union to put down an uprising by the Mujahideen turned into a full blown occupation of Afghanistan. A fully mechanized Cold War army went up against people wielding small arms and lost. (Only well into the war, 1986, did the Mujaheddin acquire modern weapons such as anti-air missiles.)
A series of conflicts over several decades between the native Maori and the British/NZ government to enforce their version of the Treaty of Waitangi. The outnumbered Maori had muskets, shotguns, and melee weapons. The British had muskets, but also artillery and a navy. Lindybeige has a good video about the Battle of Gate Pah. While the Maori decisively "lost", taking heavy casualties and losing much land, like the Finns, they continued to exist. Their spirited resistance allowed them to escape the annihilation and total subjugation that befell most native peoples.