As a trained musician, I think I have a reasonable ability to assess the skills of other musicians. Sometimes, I will watch a video of a live performance and see (in, say, an improvised solo or vocal run) a moment that strikes me as a mistake, and I suspect that the majority of trained musicians would agree. A note wasn't in tune, didn't fit with the chord, the rhythm was unclear, or the timbre was off.
But if I scroll into the comments, I'll see an argument like this:
It sounds like the guitarist got a little lost at 0:55. That note doesn't fit with the scale.
Actually, that was intentional.
But it sounds bad.
It's supposed to sound bad. That's what makes it expressive.
Indeed, out-of-tune notes, dragged/rushed phrasing, and playing "out" are valid musical devices that can have a powerful expressive effect. So, what is the difference between an expressive embellishment and a mistake?
In the case of any particular song, the line between these two will be a matter of opinion. There's no way to get inside the musician's head, so
It's not a mistake if it was intentional isn't useful. Also, people have any number of nonmusical factors (e.g. being a fan of that band) that influence how charitably they will listen to the song in question. So
It's not a mistake if it sounds good is just passing the buck.
But without touching these subjective issues, I believe we can agree on general criteria that distinguish mistakes from embellishments. Here are two examples, one crude and the other more nuanced:
It's not a mistake if the musician winks at the audience beforehand.Here, clearly, the "mistake" serves a humorous purpose. The Brad Paisley song "Make a Mistake With Me" uses this concept.
Playing the ♭5 over a minor chord in jazz is bluesy and nice; playing the ♭9 over a minor chord is generally a mistake.The melody of Jobim's "Corcovado" is an exception, but I'd still be comfortable with adopting this "rule" in the majority of cases.
Different genres have different conventions, so I welcome answers that focus on a single genre.
Additional questions to consider:
What are the nonmusical factors that affect how charitably we listen to music?
Does the concept of a mistake apply to improvised music in the first place?
Why is it useful to distinguish between mistakes and embellishments?
Can mistakes be further subcategorized?