Most people think a Battlemage is a hybrid, but its not. A Battlemage is just a mage that specializes in Destruction magic, just like a Conjurer is a Mage that focuses on Conjuration, an Illusionist is a Mage that focuses on Illusion. Not all Battlemages wear armor, but since they're mostly deployed in wars, armor seems appropriate.

There is actually a big difference between the two, even though most people assume that the " Battle" part of Battlemage translates to a Warrior/Mage. In fact the "Battle" just indicates a Mage thats skilled in "Battle" magic (Battle magic meaning Destruction). If the class was called "Destructor" or just "Destruction mage" then it would be much more clear on what it really is.

A Spellsword however, is actually a Warrior/Mage hybrid.