Is an intensional genus–differentia definition an invention of ancient greek philosopy?
"Chair is a seat typically having four legs and a back for one person"
Have you seen definitions like this before Socrates ever in human history?
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'Definition' has different meanings in philosophy and logic. No single person originated them all. But you focus on definition per genus et differentiam. So far as I know this mode of definition derives from Aristotle who uses the language of genos and diaphora. At the very least, I'm not aware of any earlier philosopher (at least in the Western tradition) who definitely and unambiguously employed this mode.
The Stanford Encyclopedia tells us :
Since a definition defines an essence, only what has an essence can be defined. What has an essence, then? That is one of the central questions of Aristotle’s metaphysics; once again, we must leave the details to another article. In general, however, it is not individuals but rather species (eidos: the word is one of those Plato uses for “Form”) that have essences. A species is defined by giving its genus (genos) and its differentia (diaphora): the genus is the kind under which the species falls, and the differentia tells what characterizes the species within that genus. As an example, human might be defined as animal (the genus) having the capacity to reason (the differentia).
To check on Aristotle directly, see Categories 3 a 33-b 9 and Topics 109 b 4-7.
Aristotle acknowledges Socrates as the originator of definitions but Socrates, whether the historical Socrates or the Socrates of Plato's dialogues, does not appear to have used the genus & differentia method on which your question focuses.