Jew, Jewish, Israelite, Israeli...
Yes. Being Arab is an ethnicity, while being Israeli is a nationality. 20% of Israelis are Arabs.
An Israeli is a citizen of the modern State of Israel. An Israelite is a person from the Biblical Era that was a descendant of Jacob and probably lived in either the United Kingdom of Israel, the Northern Kingdom of Israel, or the Kingdom of Judah. It is conventionally believed that the Israelites are the ancestors of the Jewish people, who then founded the State of Israel roughly 2000 years later. The difference is primarily time.
Citizens of the Modern State of Israel are called Israelis.Subjects of the Israelite monarchies were called Israelites.
They didn't change from being Israelites to Jews. The 12 Israelite tribes were banished to Babylon. When they finally returned to Israel, only 2 tribes were still identifiable. Jews were named after the largest of the two: Judah.
About being a Israelite
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MIzrad HaPnim in any major city
Israeli
Legally, yes. If you are Indonesian and have your sights set on an Israeli man or woman, then make sure that they don't mind marriage with non-Jews. Then, find out if their parents are okay with it. The reason being for that, is if the Israeli is into Judaism (or in the future could possibly be), then that Israeli will want to know if you will follow them into Judaism and raise your kids together that way.
Joshua certainly would have believed Moses to be an Israelite, but nobody "alleged" that Moses was an Israelite. Moses was an Israelite because of his ancestry and because of his connection to the people and he believed he was an Israelite.
He wanted to establish a stronger connection to a Jewish-Israeli identity.