I think that that dichotomy is obviously wrong. More useful dichotomy between animals and "not-animals" involves capability of locomotion, which is something that althought not strictly correct can at least be observed from outside of the system in question.
They grow bulbs where other onions have flowers: these fall over and root, and send up stalks, and it does it all over again.
That's locomotion, just, not on the time scale we're accustomed to considering. We also have a whole category for animals which don't practice locomotion, sessile.
I would say the most central distinction between animals and (plants|fungi) is that animals have a definite body shape, while the other two have indefinite body shapes.
That isn't a perfect distinction (the occasional gecko will regenerate two tails where it had one before), but it's very good, in addition to being striking / obvious.
