Finished reading Genesis and now that I'm on Exodus, I get the impression God is dealing with Israelites and not Jews. Is this because Jacob was given the name Israel, so all his descendants were called Israelites?
Yes, that's right.
The origin of "Jew" actually comes from Judah, one of the sons and tribes of Jacob/Israel. When the Kingdom of Israel splits after Solomon, the dominant tribe of the southern kingdom was Judah, and so it came to be called the Kingdom of Judah. Though there have always been descendents of all the sons and tribes of Israel, the prominence of Judah meant that "Jew" came to be used as a synecdoche for the whole of the nation of Israel after the exile.
If you search for "Jew" in an English Bible (example, NIV), you find very few hits, in contrast to "Israelites". In fact, the first appearances of the word Jew is in the book of Ezra, which is the 15th book of the Bible! And actually, in that book, the sense of the term Jew refers, to my understanding, to the Israelites of the Kingdom of Judah. The next book, Nehemiah, also contains references to the Jews, also in the context of those exiled to Babylon (which were also from the Kingdom of Judah).
a Jew of the tribe of Benjamin, named Mordecai
The Tribe of Benjamin were part of the Kingdom of Judah before the Exile. It is unclear is the rest of the references to the Jews in this book makes a reference only to those of the Kingdom of Judah.
Several books later, in Jeremiah, we find again a reference to the Jews. This book is also centered considerably on the exile of the Israelites from the Kingdom of Judah. Some reference to the Jews again seem to refer only to the Israelites of that kingdom. For instance, Jeremiah 40:11-12:
When all the Jews in Moab, Ammon, Edom and all the other countries heard that the king of Babylon had left a remnant in Judah and had appointed Gedaliah son of Ahikam, the son of Shaphan, as governor over them, they all came back to the land of Judah, to Gedaliah at Mizpah, from all the countries where they had been scattered. And they harvested an abundance of wine and summer fruit.
Again, in Jeremiah 44:26:
But hear the word of the Lord, all you Jews living in Egypt: ‘I swear by my great name,’ says the Lord, ‘that no one from Judah living anywhere in Egypt will ever again invoke my name or swear, “As surely as the Sovereign Lord lives.”
In conclusion, there are only 6 books of the Old Testament (out of at least 39 books; exact number depends on denomination) that make a reference to the Jews as a people, and in my reading, all of these are likely to refer only to the Israelites of the Kingdom of Judah. Thus, the denomination of all the Israelites as Jews might not have emanated from the Old Testament. This is in contrast with the New Testament, which was written at a time where there were no two kingdoms anymore (there was no one kingdom, actually). At that time the term Jew already designated all Israelites. The NT only reflects this reality.
PS: notice that the books of the Old Testament were not written in English but in Hebrew. This means that, even if the English translators saw fit to use the word Jew in the aforementioned verses, the original author might not have intended to use a word representing all the Israelites. Digging into the original texts might provide more insights on this, but still the context above is to me evidence enough that whatever word they used was intended to mean Israelites from the Kingdom of Judah.