That is some pretty crazy head tossing. Usually when people say head tossing is bad, it isn't so bad, but your mare's is bad. There has got to be a reason for this. If she does it more with your trainer riding ( AND let me take a moment to ask you you have permission to air the video of your trainer riding, because you are in a critique section and all photo of persons not yourself must have persmission to be here and be critiqued). It seems to have something to do with the fact that the trainer is trying to get her to take more contact. Since she won't accept it, and the trainer might also be asking her to move forward at the same time, she ends up looking compressed and holllowing out, with her hind end trailing out behind her. I noticed a couple of times the trainer appeared to kind of jerk on the reins and thought that this is not going to be helpful.
When you ride, the horse isn't on the bit, (but isn't on the bit with trainer , either). But since you are not also asking for more push from behind, the horse feels more comfortable. I thought the movement was not so bad for a horse just starting in dressage.
In my opinion, the number one thing in ALL horsemanship and especially in dressage is FORWARD. Do what needs to be done to get forward motion. I thought you had some forward going there . Eventually you want forward but also you want some contact with the bit and teaching the horse to work forward into the bit, wherein over time you will help him to move more of the forward energy into a collected and '"upward" energy. But first is Forward. Then is reaching for the bit.
I would take some time, if you have not already (sorry if I missed that) and be most certain that there is NO physical reason for the horse to fear the bit. You might try a different mouthpiece, too. Such as, if you use a single jointed snaffle, consider a double jointed like a French link? Be sure there is no dental issue and that the bit is not too fat for her mouth and sidepieces are not pinching due to bit being too narrow.
Once these are ruled out, I would work ONLY at walk for a bit and work on haveing the horse tolerate some contact with encouragement to stretch forward and downward, and give a big realease (since she loves NO contact) when she does this and let her walk with her neck down and free. Then , take up contact and follow her mouth carefully (you need to have a good following hand and a loose elbow) and see if she'll keep her head down for a bit with contact. Work on this longitudinal stretching ( forwward and down, follow it up, ask for forwared and down and on and on). Once she gets more comfortable, keep contact more and aask for a brisker walk.
Don't even do any trot until she will let you follow her mouth with the bit and not throw her head around. Don't hold you hands too low trying to bring her head down. If she brings her head up, follow her up keeping straight line from elbow to bit, even if it means your hands go up. When she lowers her head, give a release! But don't try to "pull" her down by keeping your hands overly low.