Please remember that dressage collection is intended for a horse to give a collected gait - sustained movement with much of the weight on the rear. Western collection usually refers to a shorter duration shift of weight, and the amount of weight shifted is less. It is used primarily to allow the horse to turn faster and to prepare for sudden acceleration.
Backing up doesn't teach collection, but I'm told it improves the horse's strength in muscles that will be used during western collection. I don't know personally, because I haven't tried it.
Softening vertically and horizontally, at least with the two western trainers I've taken lessons from, is not intended to "collect" the horse. It is intended to get the horse listening to the rider and responding to the bit. When my mare was started all over for her training last November (long story, not needed here), one of the early things worked on was softening vertically and horizontally from the ground. Later it was worked on mounted. The goal wasn't to collect the horse per se, but to teach her to give to the bit and not fight it, and to listen to the rider for instruction. That isn't collection, but it is a step one would need before teaching collection. And since western riding doesn't normally involve constant contact on the bit, it is again a somewhat temporary thing.
I have no desire to teach my horse collected gaits. I do need them to be able to shift their weight to the rear enough to make balanced turns and to prepare for acceleration. For that mild and limited degree of "collection", we started with flexing vertically and horizontally, and then moved to various circles. I liked using a triangle set of 3 cones. I walked the horse into the "cones of confusion" and then decided as I entered the triangle which cone we would circle, or if we were going to circle at all. That forced the horse to wait for my cue - they couldn't anticipate since I didn't know myself. We would then circle a cone, and I would check to see if the horse was balanced. Was she falling in with the shoulder? Bending her body around the turn? Where was my weight - forward or to the rear? I can't expect her to put weight to the rear if I'm not.
With practice, we increased to trotting. Since all that turning was hard work, I set up a fourth cone about 75 yards away, so we could break out of the cones of confusion and let the horse relax going to & from the far cone. Varying the size of the triangle & our speed allowed me to vary the workout intensity for the horse. Trying to get good circles helped both horse and rider with balance.
For my limited purpose, that worked well. With Mia, I had to be careful not to overdo it, since she resents endless circles. We've spent the last 6 months teaching her it is OK to go for rides out in the desert. Yesterday was her first solo trip that included cantering. And she actually remained calm and relaxed for a short canter, instead of going into a mild bolt! For her, that is big progress!
I've started making it a practice, however, at the end of each trail ride to have her canter a few circles in our small arena. The trail rides have helped enormously with her "relaxation", since much of her body tension had been mental rather than physical.
We will never work on collected gaits in the FEI sense, nor will most western riders. We normally ride with slack in the reins and with her free to choose what parts of the path or wash that gives her the best footing, although I want her ready to respond to my inputs - which are not all thru the bit. I read an article a long time ago about how 'on the bit' really meant 'on the aids' - listening & responding to the rider's cues. If so, then by that definition I want her 'on the bit'.
None of this involves headset. The softening I talk about has do do with her listening and understanding my bit cues, so that I can help her with her balance and relaxation. This picture captures our goal for a 'normal headset':
She can do a relaxed canter like that, or accelerate into a gallop with a similar position. I remember when this picture was taken, and she was very 'soft' with her head. Take a bit of slack out of a rein, and she responded right away. That is what I was taught meant she was 'soft'. And it is very hard to teach a horse not to get strung out but to 'collect' if they are not listening to the riders cues, including the reins. I gave up riding bitless because I needed more subtle cues to get my mare to 'collect'.
No idea if all this rambling helps anyone. Got up at 3 AM with sinus problems, and 3 AM isn't the best time to be posting on the Internet. If it helps, great. If not, ignore it please.