I don't know anything about the Wharton essay, but HBS question asks you to tell them about the setback. How you frame it and the consequences it had on you personally and/or professionally is completely up to you. You can easily guide the reader's perception of you by how you choose to write the essay, so the only way he/she is going to think you're a "punk" is if you make it easy to make that assumption by portraying yourself that way through your actions, thought process, and lessons learned.
Your story seems fine to me, just make sure you capture it correctly and explain what it personally meant to you, not just "I did this, I was arrogant, I didn't get the job, I learned from it, then I did even better, wow look at me I am so awesome please accept me to HBS."