Haha I bascially burn it in all the ways you all mentioned put together.
1. Obviously to end a match, but also to end a round if it's very close.
2. If I Have red axe and I can put some real fear into them by doing an easy 6000 damage or something combo.
3. I really enjoy using it on characters like Yosuke, Liz, Naoto, just to destroy a huge chunk of health.
4. I definitely use it to finish off characters before they go into awakening, or at least do as much damage into awakening as possible.
5. I'll use it if I feel like I won't be able to keep the red axe vs certain characters.
It's important to remember that your axe will recover to level 3 on it's own without even having to attack/whiff. Also that your axe will never drop below level 3 without using Moujuu-D or Brutal Impact-B. And just to make sure everyone is aware, whiffing theoretically can get you to red axe. I remember someone mistakingly saying somewhere that it only gets you to level 3 but it is possible. It takes a long time and is 1000% impractical but whiffing can save your red axe so it may be worth knowing.
Edit: Woo, just achieved S rank ^^