It seems like the TV special was the first reference point for the Fragility 1.0 name. It aired on Dec. 21, 1999, after the European dates, but before the Japan/Aus dates.

You can see the "Fragility 1.0" name referenced in the Hotline news archives:
http://www.theninhotline.net/news/archives/1999.12.html
http://www.theninhotline.net/news/archives/2000.02.html

Prior to the TV special, it was just called the Fragility tour.

So, it was named somewhat retroactively, but it's definitely official.