No a weapon does not define a job. A weapon makes you a class with its own abilities, and you have abilities from other classes that used other weapons. You play a role you decide by equipping your chosen abilities. Wanna play Gladiator (sword or dagger user) and tank? equip curing, shield, and hate skills. Want to purely DD? equip weaponskills and abilities like blindside.
Thats not to say you can't do a little of everything, and that is where the problem is for a lot of people. The role you play is chosen by you, and with so many options it drives people crazy. Can a Conjurer equip shield and hate and curing abilities and want to tank? Sure, but it will not be as effective as a gladiator because it will not have proper gear benefits and timers etc.
From here on out is all speculation and hope
With this class system I'm hoping its like this-
-rank pugilist and gladiator to 25,
-opens the thf job quest,
-Perform quest and have access to thf job,
-Play gladiator and equip thf job no you will not be able to take certain abilities from other classes, but through quests you could obtain abilities specific to thf job only.
-I personally don't see a problem with equipping jobs on any class, so long as proper penalties and bonuses are appropriated.
Red mage/gladiator would have a little magic and stronger sword strikes than red mage/thaumaturge but /thaum would have better magic. Using the red mage job would take away stuff like ancient magic, but give new red mage only abilities such as haste/refresh through quests. Red mage/pug would use pugilist abilities/weaponskills, some specific abilities from thaum and conj more than likely, and then those specific to red mage job abilities.
Remember they might not even be jobs as we know them, it could be that each weapon class will have a few jobs and theyre very generic. I don't want this to happen but it could just be like lancer gets three jobs, DD single, DD AoE, and status changer(enhancement/enfeeb). Again I would like to avoid that and hope it is classic FF jobs.