I think the second is incorrect too. DDoS is a DDoS no matter what the intent is.

I think one could argue that one. Is a DDoS a symptom? In which case the intent is irrelevant. Or is a DDoS an attack/crime? In which case it is. We kind of use it to mean both. But I think it's generally the latter. Wikipedia describes it as a "cyberattack", so actually I think intent is relevant to our (society's) current definition.