The top-like "spindulums" below fail on criterion (1) above because they don't fall over at rest -- not because their tips or the bottle cap do anything to keep them upright, but rather because their centers of mass (CMs) are
below their recessed contacts by design.
https://youtu.be/u7_azTiWxywThe precise location of a spintoy's CM along its spin axis is often unclear in practice. But 2 simple behavioral tests provide valuable info on CM location
relative to the contact:
A spintoy
with a tip and supporting surface that do nothing to keep it upright or push it over at rest has a CM
below the contact if you see it...
A. rise to the vertical on its own and return to the vertical after small perturbations in tilt, pendulum-style, as at 0:32, or
B. precess in a
retrograde direction (opposite the spin), as at 3:50.
In practice, Test A -- to be performed
without spin -- tends to work better when the spintoy's mass or CM-contact distance along the spin axis is relatively large. Make sure that the tip and supporting surface don't interact to muddy the water.
Test B -- to be performed
with spin -- turns out to be more reliable when the CM-contact distance approaches zero. Usefully, precession rate is proportional to this key distance but independent of total mass. And all precession ceases when it's
exactly zero, regardless of mass or spin rate.