Phizzy,
"Does Genesis actually identify the evil spirit as Satan ?"
Not directly. Revelation 12:9 does identify the "old serpent" as the great dragon called devil and satan. The word translated "old" according to Thayer's defintions, "1) that has been from the beginning, original, primal, old ancient" Thus, the NWT calls him the "original" serpent.
Rev 12:9 "And the great dragon was thrown down, the serpent of old who is called the devil and Satan, who deceives the whole world; he was thrown down to the earth, and his angels were thrown down with him."
Also Satan's attempts to tempt Jesus mirrored his work in the Garden showing that he repeatedly uses the same tactics.