Welcome to the Horse Forum. This is not an uncommon problem. It is not what is usually referred to as being 'cold backed'. It is usually called being 'cinchy' or 'girthy'.
Some horses start this after one single incident and then seem to just keep on doing it out of habit. I have seen horses do it when they have been tightened up too tight when first being saddled. I have seen horses do it after being saddled with a saddle that pinches or has a nail sticking out of it and then, even after having the problem corrected, they just keep doing it. I have known horses that just seemed to start doing it one day and then forever after. I really think most of the these last horses probably were tightened up too tight when first being saddled one day.
Sometimes you can get them to quit it if you start saddleing them VERY loosely and gradually tighten them up after standing for a few minutes. I have bought cinchy horses that quit by doing this. I have seen some horses that were so bad that they had to be walked around while being saddled. Some horses will stop it if they are held in hand with their butt backed in a corner to be saddled.
Once a horse has acted cinchy, NEVER saddle it in cross-ties. Always hold it in hand or ground-tie it and we always saddle it on soft ground and never on concrete or hard ground. A friend had a cinchy horse and it threw itseft over on concrete and broke its withers.
By the way, being 'cold backed' usually refers to a horse that thinks it has to buck right after being saddled. This can be with just the saddle or right after being mounted and moved forward. It, too, is a bad habit, oftentimes unintentionally taught to horses by turning them loose or longeing them with a saddle when they are too fresh or too green. Some horses 'get it out of their systems' while others learn to like it and just get practice doing it better and better.