The gist of it is that classes supposedly mix up three different things: what a character does, what a character has, and how a character acts. The examples given are:
- A Fighter fights;
- A Magic-User has spells;
- A Thief acts like a criminal.
I think this is kind of a smokescreen. In particular, a Magic-User does not "have spells", a Magic-User "uses magic": not just casting spells, but also making magic items and new spells. Choosing to overlook what a Magic-User does and calling it instead a matter of possessions owned seems like creating a problem to complain about. Similar, even if we allowed that a Thief "acts like a criminal", acting is still doing. What is "acting like a criminal"? Stealing, breaking into homes, shadowing a victim, etc. In other words, "acting like a criminal" is based on things you do.
And that, as I said, is only if we allow that the essence of the Thief class is "acting like a criminal". There is no reason why a Thief character has to be a criminal; you could re-skin a Thief as a military scout or a private detective or in many other ways. Notice that Monks get several of the Thief skills, too. Does a Monk act like a criminal? The fact that many have chosen, over the course of several editions, to force the Thief or Rogue to be a criminal specifically is not the fault of the class, but the fault of people playing or writing about the class. The Homeless Nerd might have done better by using the Paladin as an example of a class based on how a character acts; that is specifically a Fighter with bonus powers that go away if the character fails to act in a specific way.
Although thinking about classes in terms of what they do, what they have, and how they act might be a good way of brainstorming new ideas about classes, I don't think it's very fruitful as a way of defining what a class ought to be. My own thoughts about class is that it shouldn't be about what a character does or can do, but what the character chooses to do: the old "I solve problems by _____" test. Characters can do just about anything, although they may need help in some areas (Fighters with scrolls of protection and a ring of spell storing.) But they choose to address most problems with a specific, tried and true approach.