Ruin Explorer
Legend
This is absolutely correct, rules-wise, both RAW and RAI.Booming blade specifies "if the target willingly moves". Movement over which the target doesn't have a choice does not trigger the effect. Running from a failed Dissonant Whispers save is no more willing than movement by being shoved.
However, the problem is that Booming Blade is a profoundly "4th edition-style" spell in 5th edition, and the entire concept of "willing movement" vs "forced movement" is not something 5E generally thinks about, whereas it was a big deal in 4E. Looking at discussion on this, here and elsewhere, it's clear that a lot of people see "willing" vs "forced" as more "your mind does it" vs "a physical outside force acts on you" - so they see fear and confusion-based movement as "willing" movement.
And I feel like, for 5E, that's kind of probably a better distinction, even though for 4E, which was a much tighter game on the field, and where there were stronger "meta" elements, that made sense.
Booming Blade causes a particular issue both because it's a 4E-style wording, and because conceptually, it's based on you moving out of the "sheath of energy", which obviously you running away would do.