I wish I could be more sympathetic with this point of view, but the author is (as most neo-anarchists) setting up a definition of anarchy that increases in defensibility the more it departs from the real thing. In logic this is known as "special pleading," and what it does is to force debate to follow terms whose definition is crafted to lead to only one possible conclusion.
The term "anarchy" has a number of possible interpretations, but it does in fact refer to absence of government in one form or another. It is not a new idea, but many of its modern adherents are treating the field as if Bakunin and Kropotkin, et al, never existed, and that's a pity, because they're revisiting old ground and missing some hard lessons of the past.
I am an enthusiastic adherent of a smaller, less powerful central government, but most of the neo-anarchists (and that strange amalgam that names itself "anarcho-capitalist") seem to me to be attempting to retain certain features of ordered society in the absence of those aspects of organized government that maintain that order. Those aspects are necessarily coercive - government is, by definition, coercion - and are necessarily restrictive of liberty as well. Gibbon spoke to this regarding the fall of the Roman Empire: ...the establishment of orer has been gradually connected with the decay of liberty...A long period of distress and anarchy, in which empire, and arts, and riches had migrated from the banks of the Tiber, was incapable of restoring or adorning the city [Rome]; and as all that is human must retrograde if it do not advance, every successive age must have hastened the ruin of the works of antiquity.
In short, what we saw in Rome and what we are seeing in this case is that order and liberty coexist in a continuum in which there is a necessary tension, and periodic fluctuation, between the emphasis of the two. Neo-anarchists are claiming that we can enjoy the fruits of order while enjoying the fruits of liberty as well, and while I won't deny its possibility I suspect, from a historical perspective, that it's pretty unlikely. More liberty and smaller government will, IMHO, necessarily mean less order, which we could live with more easily if we had the luxury of picking and choosing which aspects of order we can most easily do without. That, unfortunately, may be a luxury that is impossible to obtain.
The idea that government causes order is bizarre. Government is the single biggest cause of disorder, by far. We had order back in the days when we had small government. We lost it when big government came in.
One small example. Roosevelt brought in AFDC in the mid-thirties. It took a single generation to destroy the black family and bring disorder to the inner cities.