These are definitely lay person terms and not the technical terminology.
Generally "out" means a bone is out of position due to muscular or tendon/ligament strain pulling it out of alignment. It may or may not involve subluxation.
Withers out, as I understand it, generally refers to the first few thoracic vertebrae being misaligned (the withers, after all, are simply the dorsal process of the vertebrae). A rib out can refer either to the actual rib, or the thoracic vertebra that the rib is attached to, which will, in turn, move the rib itself.
I would take a lot of these terms with a very large helping of salt
Horses have a very sturdy skeleton and its held in place with an equally strong muscular structure.
Yes they can break small bones reasonably easily but it takes a lot of impact to move those bigger ones - even a rib would take some force to move it and that amount of force would be as likely to break it
If a horse is consistently being ridden or just moving in an incorrect way - such as after an injury it can cause muscles to over and under develop and good physio using manipulation and massage therapy can help resolve those things over time - one example is how a horses spine can give problems if its worked in an inverted frame for a long time - but to correct that takes an equally long time of working in the right frame and won't be solved in one visit from a chiropractor bashing away at its back