David, The names Halliburton and Burton were often used interchangeably by Halliburton descendants. For example, I had an uncle named Victor Burton Hassell, after his grandmother Emelia Halliburton Ragan. So it is very probable that this is your Charles Henry Burton.
You might also check the 1870 Dickson Co Census. You say he married in the 1880's. Was this in Dickson Co? Also check Houston Co, since part of Dickson was included in Houston Co when it was formed in 1871.
If you can find the marriage record that might give you more information.