Hmm actually from what I understand, the confusion comes because back when XIII was still in development, Square said the three Fabula Nova Crystallis games would be XIII, Agito XIII and Versus XIII. Two of them ended up being Type-0 and XV which have nothing to do with Lightning's first game and then XIII-2 and LR happened.
They never really did a good job explaining it.
At the very beginning, there was an image showing us six circles. Each circle represented a FNC game. Initially, we were supposed to get six FNC games. Right now we have 5.
At first, it was three different “Final Fantasy XIII” games that would use the FNC mythos as the groundwork.
"Final Fantasy XIII" was Toriyama’s interpretation of what a Final Fantasy XIII game would be like using the FNC mythos.
"Final Fantasy Versus XIII" was Nomura’s interpretation of what a Final Fantasy XIII game would be like using the FNC mythos.
"Final Fantasy Agito XIII" was Tabata’s interpretation of what a Final Fantasy XIII game would be like using the FNC mythos.
Each director can use the FNC mythos however they want to. You might discover some similarities, but they're not part of the same universe.
Initially, the compilation was called, “Fabula Nova Crystallis: Final Fantasy XIII.”
When Final Fantasy Agito XIII name changed to Final Fantasy Type-0, they also dropped the Roman numeral in the compilation title. Now it’s called, “Fabula Nova Crystallis: Final Fantasy" and still is to this day.
From that point on, the concept of three different “Final Fantasy XIII” games was no more. It was just simply three games part of a compilation.
During development, Final Fantasy Type-0 started to go in a different direction. Chinese mythology started to became a heavy focus. The FNC mythos, while still important, was not a heavy focus anymore.
As for Final Fantasy XV… Should be self-explanatory really. For the past 8 years, the game was always treated as a numbered title. Never made sense to associate the game with the Final Fantasy XIII trilogy either. The discussion to change Final Fantasy Versus XIII to Final Fantasy XV started 1 to 2 years after the E3 2006 announcement and it became official after Final Fantasy Agito XIII became Final Fantasy Type-0 in 2011.
What's funny is that we’ll most likely get that sixth FNC game too as Final Fantasy XV will probably end up getting a sequel. If I'm not mistaken, the FNC compilation was suppose to be a 10 year plan. Looks like they're on track.