A phylogenetic definition would have to mention at least one organism,
yes; but there are other kinds of definitions, and for most of the
last 249 years, "key character"-based definitions were almost
universally used.
They were often called "definitions", but that was always wrong.
Definitions -- non-phylogenetic definitions -- do exist in traditional
nomenclature (even though they're rarely mentioned, because taken for
granted): for example, Hominidae is defined as the family to which *Homo*
belongs. There's a type (an organism or more) and a rank. These definitions
are immutable under traditional nomenclature; the diagnoses have never been,
everyone is free to change them by simply saying so in a publication.
(Witness the change of that of Mammalia from "mammary glands" to "hair" to
"secondary jaw joint" to "fusion of prootic and opisthotic" or whatever the
braincase character was that turned the isolated braincase *Adelobasileus*
into the famous first mammal.)
(Of course, such a definition is incomplete, because the ranks aren't
defined themselves, but that's another story.)