I can't speak to other rulesets but the CSI rules clearly define what they mean by "Obvious" to avoid any such confusion or arguments:
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It may seem a little ambiguous if the non-shooting player or referee accepts the shot, but the exceptions list explicitly considers these types of shots as "not obvious" and so they must always be called. Jump Shots also meet this standard and should be added to the exception list for clarity.
A couple notes I would make:
1) In 10-ball, such a non-called shot would be a legal hit and the opponent would just become the shooter. I'm not sure why the ref called it BIH in this case.
2) In 10-ball, if a non-called ball is pocketed, the opponent has the option to give the shot back. So in this case the opponent could have overridden the referee's call by simply giving the shot back.