I am entering a SR4 campaign in which I have played before with a different character, a magician.
Another player's adept died during the time I wasn't playing and now he has a magician character.

He gloated over the fact that he has almost all the combat spells, which I didn't think was possible. He said that you don't learn Flamethrower or Fireball, you learn the "Fire" spell, of which the other two are versions you can cast depending on what you want from your spell.

I quickly dismissed him, saying that's not how it works and that they are indeed different spells, but upon checking the books I found nothing to disprove his idea with.

I still think my way is the correct one, but seeing I usually play the part of the rules police in my playgroup, I need to have something to back my case.

In short, what does the books mean when they use the word "spell"?