naw, as the guy below you said:
Precisely.
My take on a Battle-Mage would be a mage that rather than being capable of using all 4 elements, chooses to specialize in just 1, in exchange to also specialize in heavy armor and melee weapons.
A Mage-Knight would have few ranged spells, and most of the time would just slice and dice the enemies with melee weapons, while having some magical proficiency. For example, being capable of imbuing your weapon to deal arcane damage, capable of polymorphing enemies into critters, teleporting short distances, creating force-fields to dampen damage, ect.