An Israelite is a biological descendant of the man Israel whose former name was Jacob, son of Isaac who was the son of Abraham. Hence, the bible uses the terminology God of Abraham, Isaac, and Jacob in referring to the one true God.
The word Jew has various meanings dependent on context. It is believed to have derived from the word Judah, one of the tribes of Israel. It is said to have been initially applied to refer to the tribe of Judah and those tribes that aligned with Judah when the nation of Israel split into the Northern and Southern Kingdoms. The name was applied to all identifiable Israelites once reconciliation occurred since some tribes not aligned with Judah are said to have been lost in the sense people from them were no longer identifiable but are said to have integrated into other tribes or nations. So for all practical purposes the word Jew should be viewed as functionally equivalent to the word Israelite unless biblical context clearly distinguishes them.
In John 4:22 Jesus speaks of salvation being of the Jews. By this Jesus means that the nation of Israel in general and the tribe of Judah in particular is the nation and tribe God chose the Messiah to be born into for the purpose of saving people from all nations and tribes of the world not just Israel and not just Judah. For if John 4:22 restricts salvation to the tribe of Judah then Israelites from all other tribes would be excluded.
So then Romans 2:28-29 become instructive in saying God is concerned about the inner person in contrast to the outer person. It is the inner that renders one’s membership in God’s chosen nation made of people from every nation especially under Christ.
For Romans 2:28-29 says: (28) For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh: (29) But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.
The words Gentile and Greeks in the bible refers to non-Jews/Israelites; they do not refer to Jews/Israelites scattered to other nations outside of Israel (John 3:16; Mark 16:15; 28:18-20; John 17:9, 20). This truth is clearly set forth in Romans 1:16 through Paul’s use of the words first and also. He did not say to the Jews first who are/were living on Israel’s land and also those Jews who were scattered elsewhere. But rather he said “Jew” which any reasonable person would take to mean all Jews no matter where they resided/resides. If there is a first then there is a second different from the first. That difference is one not being a Jew anywhere.