Uhhh.... I don't think so? I woke every crystal, and went to the Chasm and killed Ouroboros. Unless the second ending you're referring to is the secret gyroscope movie, which I did see.
There is also the ending where you rebel against Airy by having Agnes continue to channel energy into the crystal until it explodes (you don't have to stop doing it because Airy tells you to), which causes Airy to try and kill you right then and there. This is technically the false ending because you foil Airy's plans and never let Ouroboros escape, however, it's also an interesting (and still good) ending because your party stops being so naive and actually makes a decision for themselves. ---
The only fool bigger than the person who knows it all is the person who argues with him. They're all complacent sheeple. Passion fights, but reason wins.