That is exactly how I learned it, ridesapaintedpony. If I was asked, I would consider the first horse you posted a sorrel, the second a regular chesnut and this horse (along with the fresian that Haley posted) would be considered a "liver" chesnut.
Yes, that's exactly what they are. Technically, they are all "ee", meaning, they do not carry the black gene. In addition, for a horse to be sorrel/chestnut they MUST be "ee" as red is recessive. A horse that is "Ee" or "EE" will always be black.
Smrobs- That horse is actually a Morgan. :P Correct me if I'm wrong but I don't believe the park out Friesians? Also it's veerryy rare to have a red based Friesians! Btw, I think the horse you posted is a great example of what I consider liver chestnut as well!
I too call the darker shades chestnut, and the lighter/redder shades sorrel. Then if the horse has a lighter mane than the shade of it's body, I add flaxen on top. Like "Flaxen Sorrel" or "Flaxen Chestnut". Of course there no actual correct term, so it's all a matter of personal preference.