Let me preface this with a disclaimer: This is not a discussion about which height makes sense. This is purely about international recognition by the FAI.
Wikipedia says "An international committee recommended the 100 km line to the FAI, and upon adoption, it became widely accepted as the boundary to space for many purposes.", quoting this NASA source.
This news article says "Around 1960, the FAI decided to set the limit at 100 kilometers, just for the purpose of record setting flights — that any flight above that would be considered to be a spaceflight".
Information published by the FAI is also vague on the matter, as this article shows:
The approved rules became Chapter 8 of the FAI Sporting Code. Naturally, they started by setting the rule that a flight could only be considered an Astronautical flight, and then qualify for a record under Chapter 8 of the FAI Sporting Code, when that flight goes beyond the 100 Km line, i.e. the Karman Line.
No mention of a date, either.
So, somewhere between the late 50s and probably the mid 60s the FAI adopted the 100km line. But when exactly? On which date was this border adopted?
A possible avenue for answering would be tracking down two 1960s editions of the FAI Sporting Code, Section 8. I wasn't able to, but the border is defined therein.