Any difference between 'Jew', 'Israelite', and 'Hebrew'?

There are so much that I have taken for granted in term of the concern of local congregation. Take for example, yesterday during Bible Study session, the issue about the identity of 'Jews' and the 'Israelites' came up.

Are the Jews identical to Israelites? 

I have all this while assume that they are the same. Yet how sure am I that they are indeed identical? 

There was the notion among local Church members that 'Jews' refers to a 'race', while 'Israelites' is a theological synonym to 'people of God' (therefore 'Christians' are also called 'Israelites').  The former is strictly an ethnic group, while the latter is more inclusive (one can be an 'Israelite' by embracing Judaism, but one cannot be a 'Jew' by that). 

I didn't know whether was this correct. So after the Bible Study session, I did a search.

I knew that 'Israelite' refers the descendants of Jacob, who is known as Israel. I also knew that after Solomon, there were two kingdoms, with the north known as Israel while the south as Judah. But I didn't know how did the reference 'Jew' and 'Israelite' came to be, and whether is there distinction between the two.

According to Oxford Dictionary, the word 'Jew' originally used during the Middle English era, derived from "Old French juiu, via Latin from Greek Ioudaios, via Aramaic from Hebrew yĕhūḏī, from yĕhūḏāh" ('Judah').

So, 'Jew' is just a translation of 'Judah'. This means that a Jew is also an Israelite, a descendant of Jacob.

How about 'Hebrew'? 

Oxford Dictionary states that the English word 'Hebrew' is translated "from Old French Ebreu, via Latin from late Greek Hebraios, from Aramaic ‘iḇray, based on Hebrew ‘iḇrî understood to mean 'one from the other side (of the river)'."

This word is synonym to 'Jew' and 'Israelite', though it too can mean the Jewish language. This may shed some understanding of 2 Corinthians 11:22 as Paul highlighting the three different aspects of his ethnicity: "Are they Hebrews? So am I. Are they Israelites? So am I. Are they Abraham’s descendants? So am I."

He was a true Hebrew because he spoke the language. He was an Israelite because he was from the lineage of Jacob. He was Abraham's descendant because he was born into the covenantal community. This may suggest that these three aspects in Paul's understanding constitute a true Jew.

The more interesting note about Jew's own understanding of these phrases can be found at Judaism101 website. It is highlighted that the Jews see ethnicity as inherited through the mother:

The Torah does not specifically state anywhere that matrilineal descent should be used; however, there are several passages in the Torah where it is understood that the child of a Jewish woman and a non-Jewish man is a Jew, and several other passages where it is understood that the child of a non-Jewish woman and a Jewish man is not a Jew.

In Deuteronomy 7:1-5, in expressing the prohibition against intermarriage, G-d says "he [i.e., the non-Jewish male spouse] will cause your child to turn away from Me and they will worship the gods of others." No such concern is expressed about the child of a non-Jewish female spouse. From this, we infer that the child of a non-Jewish male spouse is Jewish (and can therefore be turned away from Judaism), but the child of a non-Jewish female spouse is not Jewish (and therefore turning away is not an issue).

Leviticus 24:10 speaks of the son of an Israelite woman and an Egyptian man as being "among the community of Israel" (i.e., a Jew).

On the other hand, in Ezra 10:2-3, the Jews returning to Israel vowed to put aside their non-Jewish wives and the children born to those wives. They could not have put aside those children if those children were Jews.

To sum up, 'Jew', 'Judahite', 'Hebrew', and 'Israelite' can be used interchangeably. Yet there are some occasions where I think a distinction may be helpful, for instance when one is writing something about the post-Solomon era of northern and southern kingdoms.