Agreed with Speed Racer. They are the same color, it just depends on who you are talking to and certain registries. Flaxen mane and tail can come with either.
Liver chestnut is a dark chestnut/sorrel
This would be considered a liver chestnut
I've heard people refer to them as "black chestnuts" before as well...
Genetically, chestnut, sorrel, flaxen chestnut, and liver chestnut are all exactly the same- though some breeds may be homozygous for certain shades or variations of the color. I believe your horse is a flaxen chestnut sabino.