Hang in there Eva. {ggg}.
>David wasn't a Jew?
That's correct. David was an Israelite like all of those Israelites he ruled in the Kingdom of Israel around 1,000 BC.
It was not until 933 BC the Kingdom split into North and South. The Northern 10 Tribes were later known as The Lost Tribes Of Israel after they were captured by the Assrians.
The Southern Kingdom was made up of the tribes of Judah, Benjamin and Levi. After the Southern Kingdom was captured, evacuated to Babylon and returned to Jerusalem, the MIXTURE of the peoples of those returned tribes became known as "Jews".
The word "Jew" came from Judea, where they were located when captured. The word Judea in turn came from the name Judah, after the name of the largest tribe in the Southern Kingdom. The name Judah (as in the tribe) comes from the man Judah, one of the 12 sons of Israel, but who was dead for over a thousand years before the name "Jew" was used.
>That is why the Muslims can claim Abraham as their ancestor, because he was a descendat of Shem, as was David?
They are Hebrews, and Semites, but not Israelites or (later for a few) Jews.
>So, Jews are Israelites, but all Israelites were not Jews?
You got it!