I have the exact same problem. I ride a 7 year old, 16.3 ex racehorse thoroughbred. So he is not calm like your ex army horse- he freaks out at a lot of things, and I hit the dirt a lot.
A lot.
I'm about your height but weigh less and it was actually my choice to buy a large horse. And yes, I love my occasional romp on a pony, but with high horses it just makes me feel like I can conquer the world, especially at the faster gaits or when jumping!
Anyway, my first fall was not good. Back when I rode western for a year, I lessoned on a fat, 15 year old, quarter horse. I practiced emergency dismounts at the walk and trot. But this horse was 15.2 hands and much slower then the horse I ride now.
I never actually just fell off of the western lesson horse (he did scrape me off a couple times by running into low hanging trees and the short barn door, though!)
My first fall on my thoroughbred was horrible. We were jumping little jumps, right, just at the trot. But my horse gets excited and overjumps it by two feet. I'm up in this barely-there two point, and of course lose balance. I'm trying to hang on, but he's broken into the canter from excitement and I hit the dirt hard.
I mean hard. My side is on fire and I can't move, the breath has been knocked out of me so hard. My head hit the sand but I was wearing a helmet. However, being immobolized, I watched those giant horse hooves trot past, inches from my head.
Thankfully he comes to a stop, and I clamber back on. I do some walk and trot, and then take the little jump again. He overjumps it again, but I'm prepared and it goes smoothly.
That was a bad fall. I was really banged up for a couple of days. I have fallen off of the quarter horse, a spirited pony, a giant friesian, and off my own horse more than a couple of times.
Most people's first falls aren't horrific- that generally happens once you start doing more dangerous work, obviously- like jumping, barrel racing, tight corners at fast speeds, etc.
To be honest, the fall is not that bad. It can hurt you, yes, but as long as you are riding carefully and with a helmet, you should be fine! For me, it's usually a confidence blow. It's super important to get back on, get a bit more comfortable, and then do what you fell off while doing. If you don't, you might never get back on again.
For me, it would certainly not be out of line to ask to fall on purpose. I was taught how to do so on the western lesson horse, and my trainer says she was taught moving dismounts/emergency falls as well. However, she says it didn't and doesn't help her much- when your horse gives you a half second warning before you're bucked into the air or tumble off after a jump, you often don't have time to go, "Oh, brace myself, and drop my shoulder, and tuck my head, etc."
My dad wanted me to take tumbling lessons after a particularly nasty fall of mine, but really, like said, when flying through the air off a horse, are you really going to have time to remember how to tumble? It's pretty quick- once you experience your first fall you'll understand!
But if it feels like it would help your confidence, I would ask for it and get your trainer's opinion.
To me, I don't feel like you have to wear a safety vest at the canter. A helmet should be enough. Make sure it's approved by the SEF (or something like that) and has not suffered any major falls (oftentimes, even if the helmet looks fine, the shock absorbancy is gone.)
I don't wear a safety vest at the canter or while jumping (I show jump, and do some hunt seat) and no one else at my barn or that I know does. The only rider I know who wears a safety vest is the stable owner, who is an eventer. She wears it in the cross country part. Some showjumpers (show jumping is faster and the jumps are higher than hunter/jumper) wear them, but, unlike eventing, it's not required.
I'm thinking of going into cross country with my horse, and if I do, when I go to competitions I will wear one. (I'll have to!) I won't wear one at home simply because we don't have a cross country course. Cross country is more dangerous because it's full gallop at solid obstacles, so...
The owner of the stable I board my horse (the eventer!) practices at home without a helmet even! It drives everyone else crazy, but it's her decision, and the safety equipment you wear once you turn eighteen is your decision. (Though trainers don't teach students who don't wear helmets.)
A lot of people are worried about the fall, especially people on tall horses. But usually you are just shaken up. Just wear a helmet (and your safety vest, if you want), don't ride recklessly, and learn how to emergency dismount, if you can.
Good luck, and have fun! In horseback riding it's inevitable you'll fall, and probably often- but don't let that stop you!