One of the least interesting conversations I find one can have, are discussions about whether or not anarchism is a misnomer, or whether this one act or that other act somehow disproves it as anarchism. These arguments of semantics are so disheartening; it's no wonder we can't have stateless liberty when people cannot picture a voluntary life that looks relatively similar to what they have now, with the distinct absence of violent and unjust incarceratory coercion from those with power or authority.
If you cannot imagine a leader who is not a ruler, then your experiences in life must be very narrow; leaders reveal themselves every day by doing what seems mundane but makes all the difference for their community. If your neighbor begins sweeping the sidewalk in front of their house, after seeing you out there every week doing the same, that makes you a leader, but you have exercised no rule over your neighbor; to the contrary, you showed how much you love your home and neighborhood that your neighbor couldn't help but make the same decision. When a community leader opens a boxing club, and gets in the ring with every one of his pupils, taking care to mind his strength output and take the full strength of their blows, he is leading without ruling. Those kids can give up at any time; the exercise requirements alone to keep up in boxing are very strenuous; but many of them will stick around and the relationship narrows from teacher and pupil to teacher, pupil, members of a community together.
Leaders fulfill needs; they humble themselves to the needs of others in front of them. Archons, on the other hand, I shouldn't even have to describe; if they can officially disagree that a peaceful protest is a peaceful protest, and rob and trample their own citizens without so much as a word from other nations against, then they are leading nothing. I'd argue they're following — either following their fears or following the rulers that are leaders above them.
So would leaders exist amongst anarchy? Of course! People would still respect the leading cardiologists and neurologists for their insight into their fields; people would still organize to fix what they saw as needing fixed and distribute amongst their skills and experience to accomplish the task necessary; new ideas will come to be and come in conflict with other ideas, each best espoused by only a handful of elocutionists and writers. The only difference between anarchy and the world we have now is the absence of violent and incarceratory coersion as an accepted means of leadership.
There's also a detail to this that many people like to overlook when talking about liberty; there are people who just want to follow. Ill-equipped to cover all their own needs, structured environments give them exactly what they need. Some will outgrow that need, others will come upon experiences in life which force them to adapt, but others still will find themselves in need of support and protection through their entire life. And they might be your sister, or your grandmother; they might be your father or your best friend since you were a child. It brings me disgust to see people conflate leadership with rulership. A ruler, he wields power. But a leader humbly respects the power placed in his hands and acts with honour. Certainly, a ruler is able to do the same, but would you say that he acted as a good leader in his term if he did not act with honour? Why, then, would you hold anyone else to different terms, when in such a world both of you have a sidearm on your hip anyway?
Even then, there's also a matter of scope. Let's take, for a moment, something people would think impossible: a President of the Anarchist American Region. This individual is elected as the Commander in Chief, in this mental exercise, by an electoral college vote composed of contractural representatives sent to fulfill one purpose — electing the president. In this world only ever so slightly different from ours, you have no obligation to vote for the president, but you would likewise have no vestment as a stakeholder when the Commander in Chief of the Defense Army and Defense Navy make purchasing or business decisions. Likewise, they would have no right to operate on your land or in your airspace without invitation. If you do not benefit from a service you would not pay for it, and your social network and “governance” would be limited to the limitations that exist on those within the community with which you interact and operate. And the cherry on the cake; there's an implicit understanding, being one of the families in these Anarchist Regions, that keeping a gun behind every blade of grass is not only what keeps everyone else away; it keeps our own sanctioned weapons of war turned outward and on a defensive stance.
Now, I'm not advocating for anarchy of this variety; I merely lay it out there as an example of what one person's understanding of anarchy, completely without violent coersion, could look like. More often than not, though, the scope of anarchy rarely if ever needs to exceed the bounds of a single property. Another interesting example of anarchy to look at is the matter of how English privateering ships were run and operated in the Golden Age of Piracy; while people might think of the captain as a dictator, he's only permitted to lead by an agreement of a large majority of the crew, and the crew likewise is contracted with very specific earnings, expectations, and limitations. Even the looting was remarkably fair; with the quartermaster (the logistical leader of a pirate ship) keeping close count of everything, you are sure to get your single portion of the earnings. Crews also had a very easy time of addressing a captain who was a bad leader: mutiny and heaving the asshole overboard. No one benefits from a bad leader, from someone who rules with cruelty and callousness. It's a shame the kinds of people so many are calling “leaders” these days can't be given the same treatment.
At the end of the day, anarchy is not a synonym for chaos or somehow misnamed; being such a young and underdeveloped philosophy, it is rather poorly discussed. And that's probably par for the course, since most anarchists just want to be left alone and likewise avoid bothering others. Anarchy is a lot of things to a lot of people. But the one thing it is not, is “debunked” because of midwitted contrarian semantic arguments. It is the absence of violent and incarceratory coersion, and little more. Everything else is just... part of being human.