You are describing the ethnicity, not the race. Race refers only to the skin color of the person, whereas ethnicity refers to the cultural and lineage aspect of that person.
> Each of the major divisions of humankind, having distinct physical characteristics.
Then, further down:
> a group of people sharing the same culture, history, language, etc.; an ethnic group
ETH'NICAL, adjective [Latin ethnicus; Gr. from nation from the root of G. heide, heath, woods, whence heathen. See Heathen.]
Heathen; pagan; pertaining to the gentiles or nations not converted to christianity; opposed to Jewish and Christian.
ETH'NIC, noun A heathen; a pagan.
RACE, noun [Latin radix and radius having the same original. This word coincides in origin with rod, ray, radiate, etc.]
1. The lineage of a family, or continued series of descendants from a parent who is called the stock. A race is the series of descendants indefinitely. Thus all mankind are called the race of Adam; the Israelites are of the race of Abraham and Jacob. Thus we speak of a race of kings, the race of Clovis or Charlemagne; a race of nobles, etc.
To quote the TV Show American Dad's theme song "Shining a salute to the American race." ... a line I thought would be very controversial but just isn't.
AMER'ICAN, noun A native of America; originally applied to the aboriginals, or copper-colored races, found here by the Europeans; but now applied to the descendants of Europeans born in America.
1. It's a definition for the word 'American', not the phrase 'American race'.
2. It's from 1828. I don't think usage from 190 years ago is very relevant. Language changes pretty quickly.
3. That does not match what OP was talking about. They were talking about a further distinction within white Americans. Unless I'm mistaken.