ADC vs ACC
Maybe we are oversimplifying it.
Active Distance control- This is the cars ability to keep its distance from the car in front.
ACC Adaptive cruise control- This statement doesn't mean the same thing, it indicates that the cruise control has the ability to adapt to different situations. It may be a higher function of the cruise control for example
ACC has several systems including City emergency braking, Front assist, Post collision braking, ADC, Lane assist and the speed limiter.
I'm no expert but it may explain why there are different terms.