Nowhere in Genesis does it actually say "Spiritually Die." But also nowhere in the bible do we see the word "Trinity." These are things that theologians have given a name to describe things in the bible.
The bible has a continous concern for the spiritual life as opposed to the physical. God is always concerned about what is truly in your heart. Not what you do in the physical.
Spiritual death is a penalty. It is a seperation from God. When God said "...you will die." He meant it as he will punish them.
Now the question of whether or not the serpent lied or not. No he just confused the situation. He said that they would not die(physically) and he was relying on the idea that Adam and Eve did not understand God about it being a spiritual death.
And Satan was the serpent in the Garden. He had possessed the serpent. Satan is also referred to as a serpent in Revelation.
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 (Revelation 12:9).
And he seized the dragon, that ancient serpent, who is the devil and Satan, and bound him for a thousand years, (Revelation 20:2).