Alot of people seem to disagree with what Red Mage is really supposed to be. In an effort to explain the job (and why people think it was ruined by XI) I would like to take a moment to discuss the original Red Mage, circa 1987, with it's humble beginnings as one of the 6 original jobs in Final Fantasy 1.
This original iteration of the job is, in my opinion, the best example of the job out of any of the subsequent games. This is because it is the only version that stands strong on it's own merits alone, where it is not merely a steppingstone on the way to more powerful jobs nor is it a one trick job with doublecast (which was only even considered powerful when combined with stronger abilities from other jobs).
The original Red Mage was nearly as good with a sword as warrior (and all the best weapons in the game were swords) and could use both black and white magic (though their selection was more limited then either black or white mage). Nevertheless it was a valuable asset to the party. However the play style of the Red Mage was never glamorous.
In random encounters the Red Mage was strong enough to kill most monsters in one round with a normal attack with a sword. This was great as it allowed you to easily dispatch attacking groups of 3 in a single round (assuming your 4th character was either a white or black mage) while saving precious MP for when it is most needed (since you could only recover MP at an Inn or with the rare Ether). This was usually the most effective use for Red Mage in these encounters since, even though they had access to some spells that damaged all opponents the damage was usually too weak to kill them in one hit, meaning it would take 2 rounds to kill a group of 6 mobs either way.
In Boss fights a Red Mage greatly enhanced the party's offense and survival. With access to the spells Haste and Temper the Red Mage could more then triple the damage of other characters (or his own but that was less effective since casting the spells takes up your action for the round while another character can attack while receiving the buff in the same round). Red Mage also had access to strong single target healing spells (they were restricted from using the full party heals) so they did well spot healing critically injured party members while a white mage cast heals to keep the rest of the party at full health. This was usually the most effective use of Red Mage during boss fights because usually by the time you finished casting Haste and Temper on two other melee characters either the boss was dead or your party were all injured badly enough to require the additional healing.
This is what Red Mage is supposed to be.
In my opinion the version of Red Mage in FFXI was originally meant to be a faithful reproduction of this original archetype. Unfortunately the more complex mechanics of non-turn-based games are what lead to the downfall of the Red Mage as a job that is useful in and of itself. In XI typically the most efficient way for a Red Mage to deal with a mob that was decent challenge or lower was with a sword, but because of the mechanics of the game mobs that were this weak were rarely worth killing, and all the stronger mobs end up being treated as 'boss' fights meaning the most efficient use of the red mage's skills are in buffing and healing.
Therefore, while I really like the original concept of the Red Mage as a sword wielding generalist mage, the mechanics of any non-turn-based game will render the majority of their skills useless in most situations. We have to take this opportunity to redesign the Red Mage job into one that will be more useful in a real-time, party based environment.