Before, you said you trained horses for twenty years and then retired. Now it's ten.
Cold water? Probably not, it's often given as an excuse, but reaction to cold water is very brief - usually the muscles are already sore when someone says, 'oh, cold water made him sore'. The other one is 'oh it's a cold-backed horse'. Also probably not.
Also, a horse's back can't be sore when it's wet and not sore when it's dry.
Generally it is from an ill fitted saddle, riding leaned off to one side or otherwise sitting crooked in the saddle, leaning back too much sitting leaned off to one side with hands too high (the OP was doing all of these in her picture).
Also, it can be riding too long for the fitness of the horse(probably the commonest), doing unaccustomed work, such as too much sitting trot or cantering, swapping leads on a turn or cross cantering through corners, drilling or repeating an exercise too many times, or something congenitally wrong with the horse's back, or a poorly treated or untreated injury that causes chronic pain...such as when a horse gets cast, caught under a fence, catches its hind foot in its halter when rolling or scratching, or stops and its hind legs slide due to bad footing.
Most people don't realize when a horse's back is sore. When a horse isn't kept in shape and ridden frequently, his back is generally, going to be sore, even with the best fitted saddle. It's the cause of a lot of 'bad behavior'.
Adding saddle pads as the OP did, makes the problem worse, as it causes the saddle to sit higher off the back and rock more. Adding more padding is not the answer - rest and treatment is. The horse should not be ridden until all soreness is gone and the muscles are healed. Stretching, muscle relaxants, heat and rest are good treatments. Then correct the problems with the training, saddle fit or rider position to prevent it from occuring again.
The test for sore back muscles, running the knuckles down the back in the way described, is unreliable. Many horses will flinch from it when their backs aren't sore, and it in and of itself if done too forcefully causes pain. The better way to detect back soreness is with a much lighter touch over specific muscles, and watching carefully for the way the horse's gait is changed by the back soreness.
Fighting with the horse over something it won't do is a very, very common way to sore up the horse. The horse is tense and gets more and more tense as the fight goes on. The longest muscle is the back muscle and it gets the tightest.
The OP described having a big fight with the horse in which it resisted and she had to get after the horse - this is a common way the back is sored up on a young horse.
The 'fight' actually occurs because the back is already starting to hurt, the horse is getting tired, so it starts to fight back, and the rider loses his temper and fights back more.
One has to use one's judgement. It's best not to 'get into it' with the youngsters, not by backing off of a fight, but by not putting the horse in a position where it fights the rider.
You can always tell a good, knowledgeable, experienced trainer because no matter what horse he gets on, the horse just clocks along and there are no big scenes. Very little fighting, resistance, it's very, very rare occurance with a good trainer. The trainer knows how to work the horse to get something accomplished without making a scene.
Every horse he gets on 'suddenly behaves very well', and he knows when to get off, too, before the horse is tired, hurting and resentful.
The muscles of the horse's back develop slowly. A young horse with little riding has no back muscle. Their backs have to be treated with kid gloves. And once it is sored up once - it can become a chronic problem. Seen a good many young horses ruined that way.
The key with young horses is not overdoing it, and very gradually increasing riding time and work done and difficulty of work done, over a period of many months, as the muscles gradually develop.
