Am I mistaken into believing that a Battlemage is someone who uses heavy armour/magic/1H and a spellsword uses light armour instead? Probably. Anyway you should choose your armour skill and that depends on whether you want to tank the damage or avoid it. I'd currently say that going light armour and trying to avoid damage is a safer option. Either way points in the illusion school helps for versatility and crowd control and the same goes for the conjuration school. Enchanting should also be you preferred choice of enhancement if your going with the whole magic thing.
spellsword simply refers to a 1 hand user who makes use of magic in the offhand instead of shield.
Battlemage refers to someone who only uses magic, but wears heavy armor.
Battlemages sometimes make use of bound axe occasionally but not typically as their primary form of damage (more like giving themselves time to recharge). (They use destro instead of bound bow. Bound Sword is more spell-sword)
Spellswords will wear light or no armor typically...
but a heavy armor 1hander with magic could be considered spellsword still.
There really are no concrete rules to the archetypes.... they're just ideas for gameplay concepts.