Driving is a sport where people have mostly given up reinventing the wheel when it comes to trying new ways of attaching traditional harness to a traditional vehicle because it's been done for so long that the most efficient methods have long ago been discovered and used until they became the set way to do it.
So yes, I'd say there is a correct way to use wrap straps.
I'm not sure exactly what you mean by "one loop vs. wrapping it around the tug" as I've seen new drivers call putting the wrap strap actually
through the tug itself as "wrapping" it, so I'm going to describe the right way from scratch to make sure we aren't talking across each other here.
First,
the purpose of the part: The harness system has three basic purposes- to pull the load forward, to hold the load back, and to turn the load. The wrap straps are part of the turning system and just as the tug loops keep the shafts up, the use of an overgirth or wrap straps keeps them
down.
Now you aren't trying to rope them down like a recalcitrant steer so they can't move at all, you're just preventing them from actively riding up past where they should be. I want my wrap straps adjusted so that any flex between up and down is felt as a gentle shift in pressure rather than a slam in the belly or back. I don't want vibration annoying my horse and I don't want to see the shafts able to drift freely away from my horse's sides as he turns, something which forces him to work far harder and often allows the inside shaft to poke into his shoulder or want to ride up on his neck while he almost ducks under it.
Wrap straps serve as your intermediate braking system that comes into play between when the traces slacken and the breeching engages, or in the case of driving without breeching they are your primary braking system. (In that case you really need shaft stops on your cart so that there is a firm outward projection for you to wrap your straps around so the shafts can't just slip right through your tugs on a sudden halt.) Everything in harnessing is focused on establishing a balance so the cart moves smoothly and comfortably with the horse in any direction you should choose to send him without either restricting the horse unnecessarily or slopping around.
So now we come to
How to Do That: If your cart has shaft stops (little pegs either welded to the frame on a metal cart or screwed into the side of a wooden one) you should push the shafts forward through the tugs until the peg butts up against the rear of the tug loop on each side. Not all carts have these; if yours doesn't, put the shafts through until the shaft tip is at the point of the shoulder when the tugs are hanging horizontally and set the shaft down in the tug. Do up your traces so if the horse startles forward the cart will not roll backwards out of the tug loops and crash to the ground. If you have breeching, do it up next so the cart also can't roll forward and bump into the horse's hind legs. Now take your first wrap strap and bring it up between the horse and the shaft in front of the tug loop. (The trace should be between the wrap strap and the horse.) You are going to bring the strap towards you over the top of the shaft and wrap it down and under the shaft again, bringing it up on the other side of the tug loop and behind the shaft stop if you have one. Now depending on your harness this may be all you're able to do. If you have excess strap you should wrap the strap a second time behind the tug and even a third time if you need to although if that's necessary your strap is too long. Regardless of whether you have one wrap or three, now you will take the strap forward again and tuck in under itself where it passed over the tug from front to back. Bring it down to the buckle and secure it. I will often leave the strap out of the keeper until I have done up the other side as some readjustment is often necessary. Repeat this process on the other side of the horse. You want the same number of wraps on each side if possible so spend some time figuring out how you need to do things for your particular cart and harness. After that first time you'll know how many wraps to take and what notch to tighten it to so you won't have to fuss as much.
Tighten the overgirth until it's snug but not tight. You may want to walk the horse forward a step or two to help everything settle and make sure it's as tight as you want it. I also usually pull each of the horse's forelegs up and forward to make sure there's nothing pinching but that's a personal preference. Voila! You're ready to drive.
One tip I learned on here- if you don't have shaft stops, position the tugs so they are slightly ahead of vertical when the shaft is at the point of the shoulder. Then when the horse moves off they swing into vertical instead of behind it and it gives a better picture as well as making sure the horse is not pulling from the saddle.
The Regency Miniatures harnessing site has good pictures to illustrate what I was talking about with how to wrap the straps, I hope between the two of us you've gotten the help you need.
Oh, and as far as if you need breeching for trail driving my answer would be yes. While a careful harnessing job can minimize the discomfort for the horse there's nothing quite as painful to watch as a horse whose cart is slamming them in the saddle every time they stop.
ESPECIALLY without shaft stops, you really need breeching or you're risking an accident if the shafts ever slip on a downhill and allow the cart to roll up on the horse's heels.
Leia