In any case, I like Pathfinders classifications of Extraordinary, Supernatural, and Spell Like. It feels like labeling things in a world as such help to define how that world works and people on it would view things.
Is Hawkeye's accuracy "merely" extraordinary or is it supernatural? How about things that regularly happen in Wuxia? See for example the division between the different Monk powers in PF 1e.
An alternate would be to explicitly call out when things have a power source behind them ( be it magic or psionics or ki). It feels like a world where a death stare power you can attempt once a day is very different if that power is mundane/extraordinary vs one where it is magic or psychic/supernatural. Even if the power works the same.