I'm thinking the vertical line in the closeup of your original is the cursor of your DAW software, not a loud sound spike? So your original recording did not in fact clip but when simply normalized, has lots of transients (the drums and cymbals) that are over-loud compared to the rest of the signal?
If so, it's okay, you can have the best of both worlds! I'd agree that the minor, short duration clipping you allowed is ok - ears are the best judge. However, a better approach would be hard limiting. At least that's what I would use in Audition. It allows you to boost the whole signal, but any peaks are limited to below clipping. Not sure what software that is you are running, but check if it has hard limiting and play around with it!
Peace,
Sanaka