You wouldn't be any more or less of a mage for using other magic skills.
Given that you're pretty offensive (lots of Destruction, controlling spells in Illusion, and pretty much the only offensive spells in Alteration), combined with some stealth and reconnaissance (Invisibility and Detect Life), I'd say you're something of a warlock. I mean, that is, if it makes you feel more immersed to have a name for your class.
Anyway, I think your build is pretty cool and I might try it sometime.
If you're focusing on Stealth and Magic (which is what I think you're going for), if you throw one-handed and sneak in there, you'll have Nightblade, a class from previous ES titles.
Cool
