Detective? PI for hire? Could be well connected, good picking up clues, making connections? An old school gum shoe? a CSI? An occult detective?
Hmm.. Infiltration expert, with all the gadgets and gear? A James Bond or modern day Shinobi?
Joe Normal, a poor schmoe who stubled into Running and keeps getting dragged into runs. Provides a more human perspective on what the runners do, lots of edge to stay alive, refuses to kill (use stick n shock), still thinks magic is like the Trid says, but knows corporate wageslave speak like a natural (which they are).
The Enforcer. All muscle, but all natural. Works with intimidation and raw grit, not cyberware or magic. Skills and Edge and raw stats are the focus.
The Underworld Fixer/Boss. Running is a way to gain connections and nuyen to make a move in the Shadows. Either an ex-org crime figure or just a newbe, sees each run from the perspective of making alliances, or gaining the knowledge edge. Makes copies, notes connections, sells their services for 'favours'. Kind of a Face varient.
Actually the issue outlined is not the 'types'; its the perspective. A character is not a 'mage', they are an ex-corp researcher, a lone star dectective, an occult PI, a magical ninja, a triad enforcer, am explorer of the mysteriers of the Sixth World, a pawn of a Dragon, or more! Sounds like you are playing boring stereotypes (boring for me anyways), not actual characters. I find characters/themes etc interesting, otherwise I would be bored of the types as well. Types are for MMOs, not table top RPGs were you can explore any concept.