As I said, Satan is identified with the serpent, not as it. Some allege the Serpent is Lilith, who became a Demon. Never in the Genesis story is Satan explicitly said to be the serpent.
And in your own statement you just said multiple individuals are associated with the term Morning Star. Lucifer however has the most associations with it.
haSatan is the exact term used for this specific entity. It carries connotations of prosecution and opposition in the court sense.
Whereas Lucifer was identified pre-fall as Helel, or Shining One.