Keep in mind, Gentile is a transliteration through at least three languages.
Ethnos means a people of common language, history, culture, and religion. Gentile literally translates as, "the nations." The nations of what or whom? The root and the biblical usage of it is extremely clear. Sadly, the word is often used when inappropriate.
What they are really saying is Gentile means the lost tribes of Israel but also infers all of the tribes descended from them. Meaning the Greeks, Romans, and so on...
In which instance?
All instances.
a. specifically of descendants of Abraham, גּוֺי גָּדוֺל Genesis 12:2; compare Genesis 18:18 (both J), גּוֺיִם Genesis 17:6, הֲמוֺן גּוֺיִםGenesis 17:4,15 (all P); of Sarah גּוֺיִםGenesis 17:16 (P); of Ishmael גּוֺי Genesis 21:13, גָּוֺי גָּדוֺל Genesis 21:18 (both E), גּוֺי גָּדוֺל Genesis 17:20 (P); of Jacob גּוֺי וּקְהַל גּוֺיִם Genesis 35:11 (P), גּוֺי גָּדוֺל Genesis 46:3 (E); of Ephraim ׳מְלֹא הַגּ Genesis 48:19(J); of Moses גּוֺי גָּדוֺל Exodus 32:10 (J) compare Numbers 14:12 (J) Deuteronomy 9:14; of Jacob and Esau as two nationsGenesis 25:23 (J).
b. definitely of Israel Exodus 19:6 (קָדוֺשׁ׳גּ) Exodus 33:13 (both J E), Deuteronomy 4:6 (הַגּוֺי הַגָּדוֺל הַזֶּה, said by heathen compare Deuteronomy 4:7; Deuteronomy 4:8) see also Deuteronomy 4:34, Deuteronomy 26:5 compare Psalm 33:12; Psalm 83:5 (said by enemies) Jeremiah 31:36; Jeremiah 33:24; Ezekiel 37:22; in narrative Joshua 3:17; Joshua 4:1; Joshua 5:8 (JE), Joshua 5:6 (D), Joshua 10:13 (in poetry, no article); of Israel and Judah as two nations Ezekiel 35:10 (said by heathen) Ezekiel 37:22; of Judah Isaiah 26:2,15 (twice in verse); compare Isaiah 58:2; Isaiah 60:22; Micah 4:7; once my peopleZephaniah 2:9 ("" עַם); thy people Psalm 106:5 (i.e. of ׳י), compare also Ezekiel 36:13,14 (read Kt); especially of Israel and (or) Judah as sinful, rebellious Deuteronomy 32:28; Judges 2:20; Isaiah 1:4; Isaiah 10:6; Jeremiah 5:9,29; Jeremiah 7:28; Jeremiah 9:8; Jeremiah 12:17; Ezekiel 2:3 (strike out ᵐ5 Co) Haggai 2:14; Malachi 3:9. — Note. This definite reference to Israel and (or) Judah is comparatively rare; in Hexateuch not P (yet see Genesis 17:4,5,6,16; Genesis 35:11 P); seldom in exile & post-exile prophets; not Chronicles —
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