I consider a proper top one that falls when not spinning, or a least one that would fall if it didn't have extra unnecessary constrains (like the feet of many plunge tops).
This one would still qualify if you spin it upside down. Iacopo's are tops by the slightest margin.
I think that's the most practical solution for our purposes. Basically, a bicycle with detachable training wheels is still a bicycle.
The others we can call top look-alike.
Or maybe a single-word generic term like "spinner".
What about "spinner" for any device that entertains in large part by spinning rapidly about some designated axis? Examples would include tops, fidget spinners, bicycle wheel "gyroscopes", playground turntables, and even bamboo copters. Practically speaking, the spin would have to be about the spinner's greatest or least principal axis of inertia to be useful.
A "top" could be a gravitationally unstable spinner (i) supported only at a single point (tip) below its CM, and (ii) kept from falling over only by spinning on its tip faster than a certain speed. At rest, it would topple immediately.
Some additional spinner categories alongside the top category:
1. Spinner with one-point support that achieves gravitational stability (and therefore doesn't fall over) simply by virtue of having its CM below its tip. Technically, this is a conical pendulum with spin. Iacopo has some beautiful examples.
2. Spinner that's gravitationally stable (doesn't fall over) by virtue of having multiple points of support. Don made a LEGO prototype with 3 points of rolling support two decades ago (see
here). Below is a 4-point spinner I came to independently in 2015.
Still don't know what to call Category 1, but I'm liking "rolling Olney" for spinners like this one. Gould might call some Category 2 spinners "supported tops".