A good illustration of the Definition Problem is the frequent disagreements on Myers-Briggs-related discussion lists about type guesses of famous people.
If you say that Bill Nye, the Science Guy is an ENTJ and I say he's an ENFP, do we disagree? Not necessarily. Different people often understand the same four-letter codes to mean radically different things.
Many people use the four-letter codes to stand for "personality types": kinds of personalities that they've found among real people that seem very similar, suggesting that their differences are just minor variations on an underlying theme. For example, when some people say ENTJ, they mean "people like Mr. Burns on The Simpsons or T. Herman Zweibel of The Onion: cold-blooded tyrants". Other people, though, have different protypical examples for ENTJ: people like Phil Donahue and Kent Beck: people interested in fairly mediating between opposing ideas, especially when the stakes are high and tangible. Still others think of people like The Little Prince, based on someone they know in person who's come out ENTJ on a type test.
Consequently when people "disagree" about whether someone is an ENTJ, they usually don't disagree at all. They're talking past each other. The person who thinks of cold-blooded tyrants would disagree strongly that Bill Nye, the Science Guy is an ENTJ--by which they mean, "Bill Nye is not a cold-blooded tyrant." The person who uses The Little Prince as his model ENTJ might agree, thinking of Bill Nye's playfulness. But the person who first said that Bill Nye is ENTJ might have something else in mind entirely.
The reason the "people who are similar to these example people" doesn't work is because people are so complex and varied that given any two people, you can always find profound, interesting ways in which they are similar and profound, interesting ways in which they're different. To have a conversation, though, you need to be talking about the same ways in which people are similar and different. And that requires some effort: you have to communicate to the other person what distinction you have in mind, and the other person has to listen and make sure they understand you. Then you might genuinely agree or disagree.