Why does Bud always introduce himself as "Bud, not Buddy?"

Bud answers this question for readers in chapter five of Bud, Not Buddy. During this chapter, the narrative flashes back to a time when Bud's mother was still alive.  Bud and his mother are having a conversation with each other, and she clearly explains to Bud why he is not "Buddy."  She tells Bud that if she wanted her son called "Buddy," she would have named him exactly that.  She then goes on to tell Bud that "Buddy" is a name that people give their dogs.  Her son will not be called a dog's name.  

She'd tell me, "Especially don't you ever let anyone call you Buddy, I may have some problems but being stupid isn't one of them. I would've added that dy onto the end of your name if I intended for it to be there. I knew what I was doing, Buddy is a dog's name or a name that someone's going to use on you if they're being false-friendly. Your name is Bud, period."

Bud's mother gives a second reason for naming her son Bud.  She tells Bud that a "bud" is a waiting flower.   It is waiting to show the world its purpose and beauty.  To Bud's mother, her son is like that flower bud.  He's waiting to show the world how awesome he will be.  

"A bud is a flower-to-be. A flower-in-waiting. Waiting for just the right warmth and care to open up. It's a little fist of love waiting to unfold and be seen by the world. And that's you."

Bud tells people that his name is "Bud, not Buddy" as a way to honor his mother's wishes.  She gave him that name, and she was very clear that was what she wanted her son's name to be.  Bud makes it equally clear to other people what his name is and what he wants to be called.