Awsome tips. Im curious how does the stomach contents ruin the meat? If even a little bit splashes out cant you just give it a whip? I will look at some more videos thanks!
The digestive enzymes and bacteria, if left on the meat, can cause problems.
In my experience, less problems than it is usually made out to be.
Between a couple bad shots and a few experiences with helping others in similar circumstances, I can say that the general handling of your meat will affect the flavor a lot more than if it was gut shot.
Wash the body cavity as son as possible with clean water. I usually carry a 2 liter pop bottle or two in my truck or car, along with at least a wad of paper towels or better, the blue shop towels from a roll.
When I field dress a deer, I try to get them on their back, head uphill. I basically unzip the deer, cutting through only skin. Usually, then, I do the rib cage, then the pelvic bone (and crack it by opening the legs as wide as they will go to open the cut).
Then I cut the windpipe free, use it as a handle, and start pulling towards the tail end. The contents of the chest cavity will usually come free, then cut around the diaphragm. It's the separating membrane between the guts and the 'important' organs, the heart,lungs, liver, etc., which should be a puddle of blood about this point, if you hit it well).
Then carry on rolling the whole lot out and over the crotch and tail.
If you follow this, by this point, you should have a deer with a empty body cavity, attached only at the rectum, which you can now cut free, sort out the bits of guts you are going to keep (usually liver and heart), and you can get on with wiping out or rinsing out the cavity.
If the deer was gut shot, there will be a bunch of bits of deer food floating around inside. Whatever. Rinse or wipe them out. If you propped the deer up on a slope, you should be able to start at the head end of the rib cage and clean your way to the back, with any water used to rinse, running ahead of you to drain over the tail and out.
I like the blue shop towels because they can be used to sop out any puddles, wrung out, and reused again and again, so you don't need to bring a whole roll.
Before it starts to dry, you should have a nice clean looking body cavity, use the last bit of water (after you feed yourself a slug or two to stay hydrated) to rinse the last of the blood out of the tail (just to keep it from getting crusty.
I have come out of the bush with a clean inside deer, with as little as two shop towels and a couple splashes of drinking water.
The inside of the body cavity will dry and the membrane that forms is essentially, not gonna end up on your plate.
It's really not too tough to do. A little bit gross, if you are squeamish, but, whatever.
When opening up your first one, pay attention to the ### organs or teats. If you are going to quarter or halve your carcass for getting it out of the bush, the tag usually gets attached to the same bit that has evidence of species (the tail) and ### (the ### organs or parts thereof). The head may be required (evidence that you got enough, or few enough, points on the antlers)
I don't usually skin until they are home and hanging. The hide keeps the good bits clean until then.
Equipment to consider.
I like a cheap cargo tie-down to haul out with. Just a 1 inch wide one with two hooks and a quick release buckle, rather than a ratchet. One end to bundle head and front legs together, the other to form a loop to put around yer hips (not shoulders unless you really want to wreck yourself).
A decent knife. I like a small blade, it keeps the cutting in close to my hand. Long blades I find tiring to cut with and harder to control. I want a blade that I can hold and p[lace my index finger out along the top so I can feel exactly where the tip is, so when I an cutting blind, I can at least know exactly where the cutting edge is. YMMV.
A small sharpener. I like the Eze-Lap diamond files. Light and easy to use, a fine or medium. About $10 at various places.
A couple plastic bags. For the heart and liver, if you are taking them home (hey, a raven's gotta eat too!) Some folks don't, others do. I don't particularly like deer liver, but... Also for the used paper towels.
Some water.
Some paper towels.
A Head-light flashlight. Some spare batteries. Doesn't need to be fancy. I gutted a moose with a mini MagLite in my mouth. My jaw was sore for about three days. That sucked!

A folding saw. I carry(usually) a small Japanese Pruning saw, that I bought at Lee Valley, good for cutting bone (pelvis and ribs) easier than blowing through them with a knife (which can be done, too).
Cheers
Trev