I don't think that the examples that you have with the waitress and the mechanic are good comparisons, because they are "cute" after solving your problem. More accurate examples would be the waitress or mechanic saying "Oops, sowwy! Something went wrong." and leaving, without solving your problem.
I'm also on the side of being professional while the problem still stands, because the user is most probably annoyed at the situation, and wants you to take it seriously. You can be cute after the issue has been resolved.
So you would rather them inform you of their mistake deadpanned without any show of emotion?
You don't like when the waitress circles the refund on your check, and writes sorry for the trouble hope you give us another chance with a smiley face? Or when the mechanic says he'll have his body guy fix the scratch before making a crass joke and offering you a donut?
Real life interactions are full of emotional cues, and personality. This is because it can often soften the blow of bad news. If you show me a picture of a puppy along with the error message I'm going to be a little less annoyed. That said, definitely show as detailed of an error message as you can along with that puppy.