This is FFXIV's intepretation of Red Mage, keeping the actual lore in mind. FFXIV's RDM is well made and does respect Red Mages's main characteristic of being a mage able to use both Black and White magic while still having a bit of sword usage.
It's not because it doesn't exactly fit 15-30 years old games that it's not a Red Mage.

I think that people arguing on what a RDM is should first try to undersand what FFXIV is.
The context is not "RDM", it's "FFXIV".