On the other hand, cladistics by itself can be too imprecise a way of defining things, because by a strict cladistic interpretation, humans and all other vertebrates would actually be fish.
That's partially true, in the manner of the Russian nesting dolls I described above, but doesn't have anything to with precision. It's also partially false because there is no such group as "fish." Humans belong to the clade Gnathostomata (jawed vertebrates), as do bony fish (Osteichthyes) and Cartilaginous fish (Chondrichthyes). Humans also belong to the Teleostomi, which includes bony fish and tetrapods. We also belong to Tetrapoda, which includes amphibians, reptiles, etc. We also belong to Amniota, Mammalia, Theria, Placentalia, etc.