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.
The line between the two terms is blurry, but race is certainly not strictly about skin color. Here is the definition of race in my computer's dictionary:
> 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
If you look up the definitions of those words using the 1828 dictionary, you get something very interesting.
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.
Which is more complicated by the US concept of an "american race", something that isn't even seen as racial in term of skin color. It is used to draw a line between those descendants of the stock "real" Americans.
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.
FWIW, I have never encountered the concept of the "American race". Are you sure it's actually a thing? To me the line from American Dad's theme song seems like a joke (since America has many races).
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.
> 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