Not to sound mean but if you don't start forming a leadership role with the new horse it will start to behave overtime the same way the other horse did. All horses will test you some more then others. Once they get away with one small thing they will test you more and more. Then if this horse does not work out you have created bad habits for the owner of the horse has to address. It is easier to form the leadership role when you and the horse are getting to know each other. Then after the horse thinks that he is boss over you. I think every ride is a training ride. Weather it is moving cows or trail riding. All my horse do what I ask to the best of there ability. While riding you find out what you need to work on. While riding down the trail make the horse move each part of its body. This way the horse has to listen to you and does not zone out. Make him back down the trail, side pass, move the forehand, hunches, make him side pass into a gate to open it while saddled.
I do not feed my horse treats for behavior issues. I hate when a horse rips off your pants pocket or takes your hide off, because it is looking for treats. I make the right thing easy and the wrong thing hard. If the horse does the wrong thing make that behavior hard by moving its feet, but when he does the right thing reward him by making that behavior easy.
Gain the horses trust by moving his feet and each part of his body on the ground first. Then do the same while in the saddle. When lunging I do not lunge to wear out a horse. I lunge to make it think. I do lot of direction changes, lot of transition changes from walk, trot, to lope, and back all from the lung line.
To desensitize. I want my horse to THINK not react. I swing ropes, tarps and anything I can think of to that might make my horse nervous. I have seen a lot of people that try to avoid what makes there horse nervous. I personally want to find things that makes my horse nervous and confront them. When doing this I want my horse to stand still, thinking and be calm. I continue doing whatever it is that makes my horse nervous tell it starts to relax and stands still and is THINKING. If you stop before the horse stands still and is THINKING you have released the pressure and have just rewarded that behavior. It is kind of a art to know when you need to keep the pressure on and when to release the pressure. I look for the smallest changes.
Go onto you tube and search Clinton Anderson there is a few demoes to see if you like his training style. It has worked wonders on our horses. However don't look at just one persons training style. We have took lots of different training styles and made our own that works for us.