Understanding Immortality in Greek Mythology

Perhaps I'm making a big deal out of nothing, but this question has been bugging me for a few days now, so I thought I'd pose my quandary to others to get their thoughts.

I'm writing a novel using Greek Mythology. I was writing a line about what makes someone immortal, when I realized I don't know. Throughout Greek Mythology there seems to be no clear answers. For example, some mortals could be made immortal by great feats, such as Heracles, who was man but became a god. (In some texts he is set on fire, and all that is mortal in him burns away, leaving only the godly parts). 

CC BY 2.5 Paul Stevenson
Here is Ganymede and Zeus in the form of the eagle. Many consider Ganymede to be a parallel to the soul, rising to the heavens

Another example would be Ganymede, who Zeus found so beautiful he swooped down from the heavens in the form of an eagle, gave him ambrosia so he would live forever, and made him the cup bearer to the gods. Clearly, it's possible for a mortal to be made immortal. But what about another option? Is it possible to be from an immortal and be mortal? The answer is decidedly yes. For example, Linus --a son of Apollo-- was killed by Heracles, as was The Nemean Lion, a child of Zeus. Orion, son of Zeus was killed and placed in the heavens. Medusa is the most annoying to me, though. Medusa had two immortal parents (Phorcys and Cleto) and two immortal sisters, but yet, most texts consider her mortal. In some Greek Myths (understand most Greek Myths have variations) Perseus beheads Medusa, using his shield to see her so as to avoiding looking into her eyes. Normally decapitation would kill just about anything. However, one of the things that nags at me is that Medusa's head, which was said to be affixed to Athena's shield, still turned people into stone. What? How on Earth would that work? If she's dead, her eyes should no longer be magical? She's dead! All the magic should just drain away, right? I think there is a good argument to be made that Medusa isn't mortal, she's merely in two pieces. If someone were to, say, put her back together (that is, if she existed in the first place) she could come back to life. Not that anyone would want that.

CC BY 2.5 robin.elaine
Perseus beheading Medusa

As I dug deeper into Greek Mythology, and who could and could not die, I asked a friend what he thought. He postulated that gods, goddesses, demi-gods, etc... were immortal until someone found a way to kill them. For example, vampires, although not in Greek Mythology, are often considered immortal, yet they can be killed with a stake through the heart. While I understand the rational, somehow this seems silly to me. It's like saying we're immortal until we die. I guess I don't view vampires as immoral just more Death-resistant than humans.

I understand that this is just idle speculation, and in the end none of this matters. Accept to me, who is trying to build a world in my novel that makes some sense.

Tell me what you think. Do you think that an immortal being could be killed? Or, like me, do you think that if someone is truly immortal they could never die?