TL;DR: ATC says they gave JAL Airbus clearance to land and told the Coast Guard Dash 8 to hold at the threshold. The Dash 8 pilot says they were given clearance to take off. (Procedurally I think takeoff clearance happens after the aircraft is given clearance to enter the runway, but that's how the article is written. Also I'm not a pilot)
My speculation follows: Most likely IMO the ATC is correct, the Airbus was given clearance and the Dash 8 was instructed to hold. On the safety side either the Dash 8 pilot flying did not repeat back the instructions given, or they did and the ATC didn't realize their instructions were misheard. Less likely but possible is the ATC gave conflicting instructions to both aircraft.
As with any investigation the point is not to place blame, but to see what safety measures failed and what mitigating factors could be adopted to reduce future incidents.