I agree with the content of the link, but not with what you have posted. Your definition is more relevant to Cat C. A Cat D is when the cost of repair is probably lower than the car value.*
As for comments about insurance companies not knowing about whether they need an engineer's report or not, that is usually a Policy matter. (Their Operating Policy, not Insurance Policy.) My last car and current car were both Cat Cs, and my insurer is quite happy without one.
*Cost to insurance company for recovery and storage, for instance, could be enough for the insurance company to write a car off as a Cat D, or cost of a courtesy car whilst parts are being obtained.