I think you don't fully understand the difference between having lore and a game being driven by its story. Ill use some simple examples for you.
World of Warcraft. It is an MMO with an incredibly rich lore.
FFXIV. Also an MMO. There is also a ton of lore for the game.
The key differences that make XIV Story Driven and WoW not lies in the delivery of that story and how relevant completing that story is to your ability to play the game in its entirety.
In WoW, you can start a brand new character, no life it to level cap and immediately dive into raids of the highest difficulty all without even doing a single quest with some very specific racial exceptions like Pandaren.
Now in FFXIV, you cannot raid without completing the MSQ. You will be denied access entirely unless you complete the story. Now, completion is not the same as paying attention to nor caring about the story, but you absolutely have to complete those quests in order to function. Hell, your job will barely function properly if you don't do the required Job Story Quests.
Furthermore, in FFXIV as a brand new character, you will be taken on a full journey from ARR through HW and Stormblood up to the current expansion. Boost potions excluded, you need to play through every expansions story in sequence in order to reach endgame. Each Expansion itself is a continuation of a single story that begins at level 1.
In WoW, every expansion that is not the current one takes place in its own little time bubble. They are disconnected from each other. Even though the events of WoD lead directly into the events of Legion and subsequently BfA, You can literally play through the Legion storyline without playing WoDs first.
I cannot think of any other MMOs that push the story so heavily that every activity we do is tied directly to it the way XIV does.
TL;DR, WoW has a story if you go looking for it. In FFXIV, you don't have a choice, the Story is only reason you have access to content at all.