What's the difference between "Jew", "Israelite" and "Judaizer"? Can these three terms be used interchangeably? Can only one or only two of these terms be applied to one person?
There are several, sometimes overlapping families of terms in play:
This name was given to Abraham's grandson, Jacob, by God and means "God prevails" or "God fights". There's some disagreement over which meaning is accurate, but the source is from this passage in the Torah:
Since Jacob fathered the twelve men who are the founders of the 12 tribes, his name was often used as shorthand for the entire community. We see the same phenomena with Moab, Edom, etc. Sometimes the nation was called Jacob, but Israel is more common. There are several variations:
Judah was one of Jacob's sons and the ancestor of David. As such, the tribe of Judah acquired significant political and cultural importance. When the kingdom of Israel split, the southern portion was called Judah even though it included Benjamin and portions of other tribes as well. Like Israel, there are several variations of this name as well:
Although not mentioned in the question, "Hebrews" is a commonly used term for the ancestors of Jacob while they were living in Egypt. The name may have derived from Abraham's ancestor, Eber. Other etymologies have been proposed. The word generally signifies the language of the same name, but at various points in history it has been identified with its speakers. The term emphasizes the text, language, and culture of the people rather than their nation or religion.
One of Noah's sons, who is identified by Genesis as an ancestor of Abraham. Many people groups fall under the banner of Semitic, but because the term antisemitism refers to an attitude of hostility to Jews, the word Semite is sometimes mistakenly used for Jews alone.
While there is significant overlap between the term Israelite and Jew, these terms are not always interchangeable. "Judaizer" probably should not be used at all outside of specific domains.
As stated, Israelite is a anglicized version of a word meaning "of the nation of Israel" which referred to people in biblical and post-biblical times. Once the kingdom split after the reign of Solomon, that term which would have applied to the Northern Kingdom of Israel, would not apply to the Southern Kingdom of Judah. The name "Judah" (transliterated better as "Yehudah") is what became "Yehudi"= which we got (through the German I would guess) as "Jew." Of the three it is the only one in use now.
A Judaizer is one who gets others to convert to Judaism or, separately, someone who adopts practices to live in a state which conforms to certain Jewish norms without being a Jew. It is not a term used for Jews or generally, by Jews.
The first two can be used interchangeably, although the use "Israelite" is slightly antiquated now, and is used mostly to refer to the Children of Israel in biblical times. Nowadays, the only term of those three that are used to refer to Jews is, "Jew"
As far as Judaizer, I have come across the term before, but I had to look it up, because it's so uncommon.
The Wikipedia entry on "Judaizer" defines several different uses for the term, none of which would be synonymous with "Jew".
Israelite is the term for the nation of Israel in ancient times as depicted in the Bible. Jew is the modern term for practitioners of Judaism. It would be anachronistic to describe the ancients as 'Jews' or modern members of Judaism 'Israelites'. I have never heard the phrase 'judaizer' before.