@MichaelEngelhardt I concur: in comparison to force the concept of energy is a mathematical instrument rather than physics content. To find the potential the force (as a function of spatial coordinate) is integrated over distance. The potential as a function of spatial coordinate is well-defined only when the constraint is holonomic. The takeaway, I guess, is that trying to impose absolute demarcation is ill-conceived. Rather, I suppose, one should think in terms of difference in level of abstraction. Force. Energy. Action.

I tried the throwing-a-book-to-spin-around-the-intermediate-axis. (Hardcover book, elastic band around it so it wouldn't fly open.) It's fairly easy to make that throw accurate enough so that it remains below the threshold. At the same time, it is also easy to give just the right bias to make it do that mid-air 180. As you point out, that 180 is not a discrete flip, it's a hilltop-to-hilltop type of motion. I wonder: when the object does the 180, what is the trajectory of m with respect to inertial space? Is it (to an acceptable approximation) a semi-circle?

When writing about rotations in space axis names I borrow axis names from aviation: rolling, pitching, yawing (for instance in my 2012 answer about gyroscopic precession ). I'm now thinking about a reformulation of the intermediate-axis-phenomenon in terms of motion with respect to the inertial coordinate system. The parallel: in both cases the explanation hinges on the overall object's rigidity.