I think a couple of things need to be clarified. I don't think the terminology has been officially agreed upon so these are my estimations of what is going on:

A chamber insert is a piece that converts a firearm's chamber, either permanently of semi-permanently, to fire a similar sized cartridge. The most common, and possibly the most successful is the small ring that allows a .308 cartridge to be fired out of a rifle designed around the 30-06 cartridge. How successful this device is depends on how well it is made and if you desire it to be converted back or permanent. The best, most reliable, are those very tightly machined of quality steel that are designed to be installed once and stay. These may incorporate an interference fit that locks them in permanently. Others are designed to be removed.

There are a slew of other that operate on the same principle of allowing a firearm designed for a larger shell to fire a smaller on as long as both have relatively the came bore size.

http://www.mcace.com/adapters.htm


The second item is a sub-caliber sleeve. This is essentially a small rifle barrel. A chamber designed to take a cartridge and a short barrel of six or eight inches is typical. These allow break-action, single shot, large-bore center-fire rifles and shotguns to fire a smaller cartridge. Generally the barrel assembly is centered in the bore of the shotgun with a spacer of O-ring.

http://www.mcace.com/shotguninserts.htm

http://www.mcace.com/rifleinserts.html

The third is the cartridge adapter. This is a hollow shell of a larger cartridge that holds a smaller cartridge. This hollow shell typically allows the smaller shell to be loaded into a magazine, handled by the firearm and, perhaps, ejected. The adapter shell is then typically retrieved and reused.

The most common seems to be a shell adapter that allows a firearm designed to shoot 5.56mm cartridges to fire .22LR cartridges. I have read that some firearms can be cycled by the .22LR shell. Most ARs seem to require manually working the action. Accuracy is reported to be good as both the 5.56mm and .22 use the same bore diameter.

There are also ones that allow smaller bore size shotgun shells to fire out of a larger bore. Reports are that there is less loss of velocity than might be expected but that groups are variable.

There are also shell adapters that allow a larger shell to fire a smaller one that doesn't share bore diameter. Like a .22LR in a 30-06. Some of these seem to have a short, all of half an inch, barrel stub. I would expect that accuracy would be quite poor but one report was that it was surprisingly good and acceptable for short range hunting of small game and plinking.

http://store.dinaarms.com/product_p/12ga22lr.htm

There is necessarily a good bit of overlap in terminology. Is a cartridge adapter that has a stub barrel a sub-caliber sleeve?