My personal opinion of waist training is that it is just a girdle. I do not know if it really causes organs to be moved around or if it somehow tells your body to burn the fat off in that area. I can see the theory about hiw it could possibly reshape your body, but that comes from feet and head bounding and neck stretching at a young age. I would love to see clinical trials with some scientific proof from an accredited institute.
There are different degrees of waist training. Some are far more extreme than others, but they're all variations on the same basic procedure.
You can actually see a very mild form of waist training being practiced every single day. Ever see a chick wearing jeans that are way too tight in the waist, and she's got a muffin top popping out? Guess what that is. It's waist training. If she does it consistently enough for long enough, it causes her body to grow that way. Her fat deposits will all occur above the waist line of her jeans because that's where her clothing allows it to happen the easiest.
Lycan, you're absolutely right. The basic premise of waist training for fitness types is using a variation of a girdle to change your body shape. What you're doing is putting some tension around the midsection in order to shorten the Transverse Abdominis muscle. The Transverse Abdominis is also called the "corset muscle". By applying constant pressure around the circumference of the waist, one relieves the tension on the TA muscles and allows them to become shorter circumferentially. This contributes to a smaller, tighter waist line. Here's a little Wikipedia reading on the function of the TA muscles:
https://en.m.wikipedia.org/wiki/Transverse_abdominal_muscle
We're not going to extremes and trying to move our organs around, although when you see women with a 20" waist and 36" hips, that's exactly what's happening. The problem with going that small is that is actually causes atrophy of the abdominal muscles to the point that these women actually require the corset or waist training device to be on for almost the entire day in order to have any core support. Anything can be taken too far. No one is suggesting that you go that far, quite the opposite actually.
Coincidentally, abdominal vacuum exercises are another great way to achieve a smaller, tighter waist line. I'll go as far as to say that if bodybuilding competitions still included waist vacuums as a standard pose we wouldn't have the current state of bloated distended guts that you commonly see on stages at the highest level.
So if anyone wants a prescription for a tight waist, it's cardio, Tren, Anavar, low Test, core work including vacuum poses, and a Squeam. Good genetics also helps. Good luck.