The sense of confusion about the definitions of race and ethnicity is entirely appropriate, because there are no good definitions for either. The US government has definitions, used by federal agencies for various purposes, but frankly they are absurd. While I was working at the NIH, I had to use those definitions as did people who answered questionnaires about their demographics. Possible races for self definition included White (which includes all Latinos), African American, Asian and Pacific Islander, and Native American. If one selected White, then there is a choice of two ethnicities: Hispanic, or non-Hispanic.
Clearly, this is a non-scientific and non-cultural, politically driven set of definitions, but it is after all the government. The official scientific definition of race (as used in research publications, and everywhere else that race is used as a demographic variable) is self-definition. There is no objective scientific definition of race. As @Christy pointed out, racial definitions vary with culture and are wildly different in both location and time. The US has the most bizarre definition of race, which is a political, historical, economic derived definition originating from the “one drop rule” that allowed Southerners to define slaves as “Black” if they had “one drop” of African inheritance. For this and many other reasons, race has no scientific basis. The Winter 2018 issue of the ASA magazine “God and Nature” was devoted to scientific and social issues around race.
Ethnicity does indeed refer to common cultural backgrounds such as language, customs, and sometimes family relationships. It is also impossible to define precisely since almost every division of a human group can be further split, but again, self-definition generally decides ethnic categories.
As for the Bible, the ancient world, like most of the modern world, was highly conscious of inter-group differences. The revolutionary verse in Galatians 3:28 “There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus.” is one we should try to live up to, as difficult as it appears to be.