Now I'm confused. Is anybody actually certain about what happened?
The issue here was, that JAP coaches shouted "OUT" to the player in the middle of the rally, and she then stopped her stroke.
This is illegal.
It's not that nobody heard it, because it was heard by the umpire, by the Danish players, for certain by the service judge, and maybe even by the Danish coaches (even though I don't know it they just picked it up from the players).
Danish players wanted the umpire to rule according to the rules, but the umpire was apparently under the impression that the player had seen the ball as out herself, and decided to let it go (which could easily have been the case, as it was far out).
Nevertheless, the situation was that the coach interfered while play was on, and he admitted it (said he was sorry, clearly gestured that it was a mistake and he wouldn't do it again).
Now, Danish players insisted that the umpire ruled no point for the Japanese, so they called for the referee (which I think is the correct procedure by the way). Referee talked to everybody, and also decided to let it slip.
Here's when I think the Danish players made a mistake. At that point, it makes no sense to keep insisting and debating, because they know that both umpire and referee agree with them about the incident, but both decide to just warn (and not punish). Danish players should let it slide and focus on the match, instead of keep insisting here. They broke concentration and easily lost the next point. (in contrast the Japanese players handled the double hit incident more experienced, and didn't let it affect their state of mind).
I think some players keep complaining to the umpire (and referee) to "put one in the bank" - so that next time an incident happens, the umpire will tend towards evening out things. This is the norm, for all players, when looking at overhand serves. You never see an overhand serve ruling pass by without some sort of complaint, because the player want's to make sure, that the service judge will then hand out a "borderline" penalty point to the opponent, OR perhaps let the next overhand server from the same player slip by. Like it or not, that's how 95% of the pro players act. I don't agree that it pays of - most of the umpires and service judges doesn't care, and often it bring the player out of their mental focus to debate...