Once again, I look at common themes in stories and find them in Oz...
Recently, I thought of your archetype fairy tale, like the classics of Andersen, Perrault, and Grimm.
Most of these stories feature a lead female character, like Snow White, Sleeping Beauty, or the Little Mermaid. Some of these characters are active and move the plot, like the Little Mermaid going to seek her desire for humanity in the Sea Witch. Others have the plot move around them, like how Sleeping Beauty and Snow White get cursed, fall for the curse, then are eventually rescued by the prince, or some other male figure, who steals the role of protagonist from the title character.
Other stories feature the protagonist seeking their fortune. In these stories, it is usually a male character. Usually, they defeat a monster or some foe, then get rich and/or get a princess or fair maiden. (These archetypes even made it into other forms of folklore.)
There are exceptions to these archetypes, but these are your two basic fairy tales.
In many of his Oz books, Baum played with these archetypes in his characters and plots. We have characters like the Shaggy Man, Woot the Wanderer, Bungle the Glass Cat, and Button-Bright who just wander Oz seeking their fortunes. However, they never stop, they keep going: the adventure is the reward in itself.
Not many of Baum's leading females fit the "Damsel in distress" model, nor do they fall into the category of Snow White and Sleeping Beauty as I mentioned above. Dorothy and the other girls from America often spark events, take decisive action, and scarcely need to be rescued. In fact, in Rinkitink in Oz, Baum turned the "Prince saving the girl" archetype on it's head by having Zella bring Inga his shoes, and then having Dorothy help Inga out of his predicament with Kaliko.
Ozma doesn't quite match your regular fairy tale princess. First off, she's actually ruling a country, so it would seem that the term "Queen" would suit Ozma better, though she is often called "Princess Ozma" because of her apparent age. (Baum ignored this in Queen Zixi of Ix when young Bud becomes King of Noland.) In The Lost Princess of Oz, Ozma is kidnapped, but it wasn't the original intent of the kidnapper to do so, and Ozma comes upon him, though she just scolds him for his wicked action instead of taking real action. (Guess she left her fairy wand in the other room...)
Instead of having the characters being rewarded by riches or power at the end of the story, Baum would often change this. Sure enough, the Tin Woodman, Cowardly Lion, and Ozma all are rulers by the end of the second book, but it wasn't a theme Baum chose to pursue later on. He even made money worthless in Oz and jewels and precious metals and stones so common that they are used in buildings and usually left in their settings.
So, really, Oz is not your average, old-fashioned fairy tale. Rather, Baum defied the archetype of traditional fairy tales and created his own style.