Personal definitions vary, but the deciding factor seems to be Armor. To me, Spell/sword is just a combo, one in each hand. Battlemage is what I call a Warlock, or a mage specializing in combat, which can switch between dual cast at range, and 2 handed in the melee. A pure Mage is simple in theory, nothing but Magic. No armor, weapons, or sneak. Similar builds are the Nightblade, a Magic Assassin, and Witch Hunter, which combines magic with Archery, among other things. I also do a rather effective Spell/Shield Combo, because a shield makes up for wearing robes with better protection, and the bash can open up opportunities to break out of the fray. (I think that's a variation on Battlemage.)
Along with the classic choices of Altmer, and Breton, Redguard works well as most of these builds, except for Nightblade. Those are best with Dunmer, since they start with all of the requisite skills.