Well, since some people seemed to be getting huffy in that thread about there even being a distinction, another thread is a great place to discuss it.
According to registries like the AQHA and APHA, a "sorrel" is the lighter colors that tend more towards coppery or orangey, like the picture that you posted. They will often have a lighter or flaxen colored mane/tail but not always.
A "chestnut" horse is the darker, richer shade, almost a blood color and darker. They will often have manes/tails that either match their coat color or are a shade or two darker. Some even have manes/tails that appear black and make them look similar to bay horses without black legs. Though there are others that also carry the flaxen mane/tail.
Then there are what's called "liver chestnuts". Their body ranges from a very deep, dark red to what may appear to be black, though they are genetically red. Their manes/tails are commonly dark, matching their body color, though sometimes you'll see one that is a flaxen.
Sometimes, if they are dark enough, the only way to know for sure is to have them tested, but there are usually indicators, even on a very dark one. Their legs will get just a little bit lighter the closer you get to their hooves, plus, they will seem to have a reddish tinge to their "black" coat when they are out in bright sunlight.