My band director in high school, my sophomore year, told an odd little story about this word. To paraphrase:
“When I was a kid, I used to play outside in the mud. Then I’d come in and track mud all over the house. My mom would call me on it, and I’d say ‘sorry’. Eventually, she told me to stop saying that, because I didn’t mean it. She said, ‘if you were really sorry, you’d make an effort to change it’.” So he told us to say “my bad” instead.