No Sony didn't go first what?
Even your own quote says the Gamecube was $199 at the start, it cut by $20 or $20 early 2002, and were already going to cut the price at $149 at E3 2002, that was planned ahead of the PS2 announcement. GameCube was always cutting prices until 2004 almost every 6 months.
So the only "others" would be Microsoft.
But insiders were already spreading rumors of Xbox cutting the price to $199 before Sony suddenly cut cut the PS2's price.
http://www.nintendoworldreport.com/news/7363/sony-cuts-ps2-price-to-199
So the answer is none of them.
What's more is that the PS2 fast price cut was in response to the Xbox to prevent it from catching up. Even though the gap narrowed later that year for a short time.
https://news.microsoft.com/2002/04/...-for-xbox-tops-1-million-mark-in-record-time/
https://www.namibian.com.na/archive19982004/2002/January/techtalk/0238195B82.html
https://www.alamy.com/microsoft-ann...-bolantemicrosoftho-apbhk-image380782525.html
Xbox had a very strong first year in NA, Sony was smart to cut the price, even if they still would have been way ahead WW, they didn't want things to be competitive in the US, although it still happened for a few years anyway, but Xbox would be prevented from catching up.