Any% requires you to get from the start of the "Dragon Adventure" mode to the credits. All Stories requires to finish all of the stories. Additional stories other than the main story sagas, can be unlocked by completing addtional tasks in certain episodes of sagas. For example, Saiyan Saga, episode 1. Defeating Radiz, whose level is around 130, with Piccolo. Will give you "Fateful Brothers" an optional saga, which is meant to be filler.

Considering that he started the run using the New Story, and the timer does not offically start until the fire cutscene starts. I would consider that would a Any% run

I noticed that the current WR has weapons that is unlocked later in the game. I personally think that would be considered NG+ (since you have the weapons in the beginning of the run). Any% would be started out with a clean file, restarting the story mode using the 'New Story' option. Timing would be the same. NG+ would have all of the weapons, and can use the 'Rewind Story' option in the mission select screen.

Anyways thanks for reading this!

Hmmm. We actually never thought of that. We are going to discuss about this. Thanks for the idea!