This is definitely super late for a response, but since this link is featured on the first page of the Google Search Index, I'll go ahead and answer it for beginners.

The save system in XIII is sort of redundant but necessary. The thing is, you can save the game wherever and whenever you want to, but the game won't load your progress from that point. Instead, it'll load the last checkpoint before the save. You typically get checkpoints at the beginning of a level or whenever it flashes on screen and the amount of checkpoints per level depends upon the length and difficulty.

Now why this is important is because the game doesn't automatically keep your progress unlike other games from the era or even modern games. Instead, you have to manually save your progress every time by either quicksaving or creating a manual save.

So if you forget to save while you're playing, you'll end up losing all your progress and and have to repeat from your last save or even from the very beginning of the game just to get back to where you left off.

As for the problem posted above, the best time to quit XIII is right after a checkpoint, i.e. after a level or once you get one of the sparse ones in-between levels. Otherwise, you're gonna have to start all over from the last checkpoint again.