It's kinda both?
Like I said earlier in the thread, pre-2e Paladins were still religious, but they weren't religious as their primary thing. They were a Charlemagne/Knights of the Round Table thing, where you were a chivalrous knight of the realm and being religious (specifically Christian let's be honest) was just an expected part of that. My guess is that D&D evolved Paladins into that more predominantly religious context so that the class could have an identity meaningfully different from just a Lawful Good fighter.
The entire evolution of D&D Paladins afterwards is a separate discussion, but in the context of FFXIV the main thing is that Final Fantasy's image of a Paladin grew out of that pre-2e Paladin and sort of evolved in parallel, basically just winding up as 'a knight with some divine-looking magic' (but not actual Holy, that's still the white mage's territory... who, uhm, also isn't typically depicted as religious, but at least is sometimes).
