There's obviously a great difference between dressage horses, and eventing or jumping horses. I agree with what has been said before, but I would add that there are big morphology differences, since we don't train a dressage horse the same way as an eventing horse.
In eventing, there's more and more research of lightness, riders are looking for light horses, unlike before where they were looking for powerful horses, with strong legs and joints. That's not the case in dressage, where horses remain strong horses, with less lightness in their morphology.
For dressage, I think we want a horse who is very serene, with a good mental, which is very important in dressage, both for the rider and the horse.
We aren't looking for cold horses, because dressage requires a lot of impulsion and strength, but it has to be under control, that's why there's an impression of coldness.
It's like in eventing, but we don't ask them the same thing: eventing horses have to 'explosive' on the cross country course, whereas a dressage horse have to use all his energy to be fully connected to his rider.