Bud always introduces himself as "Bud" because that is his name.
Bud's mother named him Bud, not Buddy; therefore, Bud wants to be called what his mother named him. Most of chapter five is a flashback to when Bud's mother was still alive. A large section of the flashback contains her explanation as to why she named him Bud.
"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 mom is quite passionate that people call him Bud. In order to honor his mother's wishes, Bud makes it clear to people that he is to be called Bud, not Buddy.
Bud's mother further explains her liking of the name Bud. She explains to Bud that a "bud" is a "flower-to-be." She goes on to explain that is what Bud is. Someday he will show the world exactly how awesome and beautiful of a person that he is.
On a personal note, I love that section of the book. My name is Jonathan. Not John. John is my dad, my grandpa, and my other grandpa. I'm not joking about this next part. My mom always told me that if she wanted people to call me John, then she would have named me John. My name is Jonathan. I completely understand Bud not wanting to be called Buddy.