Poaching is about cooking something gently, until just done. This is good for chicken breasts: white meat has very little fat and connective tissue, which makes it well-suited to this. It's at its most tender when it's not overcooked. Dark meat will be fine too. As with most other methods of cooking, it just needs to reach the appropriate temperature in the center.
Slow cooking, on the other hand, is about bringing something up to temperature and cooking it there for a long period of time. (There's no temperature to reach - it'll have been hot enough for a long while.) With meat, this is good for tougher cuts, as the long cooking helps break down connective tissue and soften it up. This is amazing for something like a pork shoulder, which can be cooked for hours and hours, until it's completely falling apart. You can probably get some fraction of that effect with dark meat. White meat will be iffy.
As for your favorite topic, salt absorption, longer cooking will get you more penetration into the meat, especially if it begins to fall apart. That won't happen easily with white meat, though, so it'll be a small effect - possibly you could cook it a while then pull it apart with a fork to help. It'll be a bit better for for dark meat.
Finally, since you've posted so many questions asking basically the same thing, my two cents: the best thing to do is probably just to roast chicken (don't overcook it!), shred it, and add some kind of sauce with a decent amount of salt. Cook it a little bit longer if you like, and there you are. You can only go so far with boiling/simmering/poaching/slow cooking, no matter how much you try to optimize it, and you've already gotten the main advice in other questions: possibly tenderize it, cook it in salty liquid, and perhaps help pull it apart once it's mostly cooked.
Yes, you are slow-cooking correctly.
If you were cooking some chicken adobo then having the meat fall off the bone is intended.
Otherwise if you want to serve if while everything is still attached, you can do so anytime after you have simmered or poached the chicken (making it safe to eat), not sure it qualifies as slow cooking however. As pointed out by @Aaronut, extended cooking will break down connective tissues - it breaks down collagen.
- If your chicken is fat, then it may take longer to break down.
- You can develop a caramel crust on your chicken so that it holds longer and lock in the juices.
- tip: put the fattier part on top, so that it runs down the chicken.
Here is some information about searing; it can cause a maillard reaction which does a few interesting things:
- it will add more complex and varied flavors, by reacting the sugars, amino acid, and heat.
- caramelize stuff, e.g. onions :D
In some cases, pressure-cooking can achieve the same results as slow-cooking in less time, while making it easier for the maillard reaction to happen.
Best Answer
While both recipes will produce "cooked chicken thighs", the intended texture of the result is different. In the case of slow cooking, the temperature is meant to be high enough to render out fat and convert collagen into gelatin. The texture of the meat itself will be somewhat stringy and shreddable, with some dryness (hopefully compensated for by the fat and gelatin). In the case of the sous vide recommendation, the result is meant to be firm rather than shreddable, with more retained moisture, and no attempt to convert collagen.
For slow cooking, the long cooking time is required to effectively render the collagen. For sous vide, you basically just want to hit the target temperature, so recommended time is lower (but can be extended without affecting things much).