I can't tell if this episode's purpose was just to teach the lesson or if Pinkie Pie really is distantly related to Applejack.
There exists an apple variety called Pink Lady which is, in all likelihood, the name Pinkie Pie saw. It would best explain why she could read the first part of the name but not the second.
BUT HERE'S A WAY COOLER THEORY that has probably been posited by people who saw this episode as it aired.
This episode confirms that Applejack and Pinkie Pie are definitely related. So here's what I'm thinking:
As shown in this episode, Apples are stubborn. They fight, they bicker, and they argue over the stupidest things. My theory is that sometime in the past, there was a feud in the Apple family and some members left to start a new clan and a new farm. However, since the Apples basically have a monopoly on all things farming, this rebel clan was left with nothing to call their own. Still, they had farming in their blood, so they chose--out of stubbornness--to farm the one "crop" that the Apples wouldn't touch - rocks.
"But how did they survive if all they farmed was rocks?"
They probably ate them, just to spite the Apple family. And so for who knows how long, the rebel Apples farmed rocks. At some point during that time and the present, they disowned the Apple name. Without knowing how marriage works in Equestria (Carrot Cake and Cup Cake? That was either an arranged marriage or someone's got a maiden name) there's no way to know which of Pinkie Pie's parents has Pie as a surname, or why they would choose Pie as a surname at all.
Fast forward to present day and you've got Pinkie Pie, the result of generations of rebels. This could also explain her give-no-bucks attitude. She also has the stubbornness that her ancestors did, as shown in Party of One when Rainbow Dash tried to force her to go to Sweet Apple Acres.