Yes, Jews as well as Catholic/Orthodox Christianity understand the term "Satan" to merely refer to an "adversary" or "accuser." For instance, the official US Catholic Bible, the NABRE, refers to the "Satan" of Job as "the satan" with a detailed footnote explaining that the individual spoken of is not to be immediately identified with the "Satan" of Christianity.
As to the rest, it may surprise you that the Jewish way of viewing this story of Adam and Even is that it doesn’t end at Genesis 3:24 but at Genesis 11:9. While Jews can differ greatly in how literal or symbolic they view the details of the subject, the following is what is generally understood about the Creation story in Scripture.
The Creation story is told in a series of stages based on the cosmology the Jews shared with their Mesopotamian neighbors:
- the universe is a vast sea or abyss of waters,
- from the waters a dome is made to hold the waters back in order for the world to be made,
- the first world is erased by these waters in order to make room for our present world,
- all peoples are assigned to their proper place by tribe or people or (as in the case of the Bible) language.
The “Adam and Eve” story is the second act of the first world created. It is likely not literal in detail, though some Jews believe that the persons were. “Adam and Eve” are merely set into the Mesopotamian cosmology paradigm, but the drama has Jewish theology lessons centering around the need for Torah observance. Instead of the trees representing actual forms of botany or an act of “original sin” being described, Jews see the story as merely telling us that humans tend to act independent of their Creator. The “serpent” is not “Satan the Devil,” just a serpent that “speaks” as part of the lesson (much like the animals in Aesop’s tales speak in those moral lessons).
The world that results from Adam and Eve is made to fit into the Mesopotamian cosmology in which it was believed that the deities grew angry with the world and wiped it out by releasing the cosmological waters upon society in order to create the current world. It grows darker and darker until the Noachin flood wipes away all things (not because of the “serpent” but because Adam and Eve’s children are immoral), and the Hebrew “history” can thus be made to fit the accepted “science model” of the universe and its origins. Lastly the current world gets divided into nations by language (a dramatic device, far from being a true historical, direct action of God).
The story ends in chapter 11 with a genealogy that leads to the history of Jewish patriarch, Abraham. The setting is that God is responsible for all peoples having their unique language, cultures, and countries, and no less the Jews--whose story gets told from this point onward.
Jehovah’s Witnesses (and most literalist Christians) read the stories as separate events in history while in reality they are just one, “the” story of “how the universe (which back then was just the world) came to be” according to the accepted cosmology of the day. The Jews merely pasted their own characters and religious lessons onto the paradigm, but the complete story is Genesis 1-11:9. This make demands upon the story that do not allow for the Jehovah’s Witness interpretation of a “fall from grace” that occurred in a literal garden due to a conversation with a snake over which tree to eat from. Who wouldn’t want to eat from God’s tree, anyway? It would be better for you that any other? (And really, does God really need a tree to eat its fruit?) It’s too silly even from a pure mythological reading to allow the narrative to be reduced to Watchtower literalism.
This doesn’t mean that “Satan” is a mere Christian invention, even though the serpent in the Garden of Eden is not identified with the Devil (nor is “the satan” of Job--which in that book is merely an angel raising issues for the sake of argument). But as time passed, Judaism came to suspect that humanity was not without some challenger that it could not simply ignore.
While today some Jews see “the Devil” to be nothing more than the imperfection or dark side within each of us which we naturally attempt to overcome, Second Temple era Judaism already acknowledged a “Satan the Devil” of sorts, a “fiery hell” and even Purgatory (this Catholic notion actually comes directly and mostly unfiltered from Judaism). The Tanakh is only a static picture of past theological concepts, absent of the actual Torah influence or tradition (often called the “Oral Torah”) which was the mainstay of Jewish religion. Without this, all reading of the Tanakh is greatly incomplete.
Later redacted into the Mishnah and expanded via the Talmud, the concept of evil was sometimes personalized into dark spirits and demons not unlike those embraced by Christianity. By the time of Jesus, the idea of a “chief” or “leader” of the demons was already in place. The term “Satan” is merely a label borrowed from Hebrew as this being (if he exists) does not actually have the name given to him by humanity.