I think you're doing the right thing, trying the horse out with another trainer. At least give the horse a chance to see what she is like.
I've known some horses that flipped, and the trick is that you could probably get a lot of horses to go over backward if you had the right set of circumstances - i.e. lots of pain or pressure and the wrong handler or rider.
I was on a sweet, gentle, mellow horse that stood straight up on his hind legs when I put on a saddle I was told was his, to all appearances it fit him fine, but it caused sudden, severe pain in his back when I sat down on it. Now if I had pulled on the reins or unbalanced him, he could have gone over. And if someone had put that saddle on him again after that happened, he might have reacted even more severely. Bucking also can be caused by a very sore back and the wrong saddle, with horses that would otherwise never buck.
Since your vet has already described back pain, it makes me wonder what happened to cause it. Also, if your horse has come over on her back twice supposedly, I'd make sure you had her back checked out thoroughly in case there is damage that may cause pain when ridden. Landing on a saddle can cause damage.
I know another sweet, mellow horse that was put down because he went over backwards once. The poor horse had been kept in a stall with no exercise for months, due to a sore shoulder. The moment he walked soundly, he was taken out and worked on cattle for 6 hours. At the end of that time, he refused to go forward. The rider spurred and used the crop on him until the horse reared up and went over backward on the rider, who had the horse put down.
My point is that you don't know how severely your horse was treated, how poorly fitting the saddle was your horse was ridden in, or what techniques were used. I don't think you can say the horse is unreliable until you've had an opinion from your new trainer.