I wouldn't take anything that happened involving Lightning seriously -- in either FFXIII or FFXIV lore. She specifically says, "I probably won't remember any of this, so please remember for me" (or something to that effect). Her appearance is just fan service and I really rolled my eyes that they would mention FFXIV lore at all. I'm fine with her randomly appearing for some promotional FATE, but mixing her up in this game lore was a bad idea.

We do have the light of Hydaelyn, but Hydaelyn is not infinite, as you will discover in the Praetorium storyline. At one point she(?) says she is exhausted and can't help anymore; you're on your own to end it. You do, thankfully, but that whole situation displays that even the light of the crystal can't absolutely save you. If it could, adventurers wouldn't be needed in the first place. As summoners grow in power, so too do their Egis. Remember the main antagonist you fight in the storyline? His Egis are larger, and he has more powers. You somehow miraculously defeat him, but he is a prime example of summoner power = Egi size/definition.

As far as the detail/size... this was merely an interpretation by me, but something I think is very real now that I've seen footage of the Titan extreme fight. If you've seen the video, you'll notice that Titan's brothers/pets/minions are the same model as YOUR Egi, except much large and slightly more defined. They aren't the BRIGHT yellow pure Earth elementals like your Egi, but rather a darker rocky color like Titan himself. They also have more defined abilities. They look more like lifelike entities instead of purified elements.

But you might be saying... "Hey! Garuda's sisters have always looked just like her and they still do!" You are right. I think this was an error on the dev team's end, because the 'sister' mechanic of Garuda has been around since she was first released in 1.0 -- except it was actually slightly different (they had different vulnerabilities). I imagine that summoner wasn't even in the works at that point, so it was not thought about when they used the same model for her sisters/minions.