It's based on both. The history also includes Palestinians living there simultaneously. In fact, prior to Islam (~700 AD), they were both Semitic peoples. I don't want to say that they were Jews who changed religion, because I don't know that for a fact. But it wouldn't surprise me.

That's also inaccurate. The Jewish people date back to the Kingdoms of Israel and Judea, roughly a millennium or so BCE (they start showing up in the archeological record around 900-800 BCE). The two kingdoms were conquered by Babylonia/Assyria in 720-580 BCE - the Jewish population was forcibly rounded up and removed from the area, exiled to Babylon. In the Babylonian exile we developed a lot of the religious and cultural traditions that make up what we consider the Jewish religion. A large population of Jews returned to the area in the 300's BCE, and we were a separate "people" from the other folks living in the area (culturally, religiously, linguistically, and ethnically) by around that time...

....about a thousand years before Mohammed.

We were still there for several centuries later, after the Romans took the area. The Jewish-Roman Wars lasted around half a century, from 66 CE to 136 CE and the Bar Kokhba Revolt. We were utterly crushed, the Jews were expelled from Jerusalem (though not from the area entirely, we were still trying to retake the area as late as the seventh century):

en.wikipedia.org - Bar Kokhba revolt

At that time, the Romans (or more accurately the Byzantines) were full on Christians, and apart from occasional conquests by the Persians (Iranians), they were still in charge up until the Arab/Muslim conquest of the Levant in the mid-seventh century.

The population in the area was not an undifferentiated group of "Semites" at the time of the Muslim conquest; and Jews had become a discrete ethnos and people a millennium prior.