No no. Check the origional 動植物檢疫 webiste. Don't worry about this article in Petism. It is confusing.
I emailed to 動植物檢疫站 in Hsinchu, and I got a response. Now there's a cage for my cat in Taiwan University 檢疫站 in July. That's it. Then don't worry about anything else. Just buy a flight tickets for your cat and see you in Taiwan. :) When you and your cat arrive JKS airport, you need to go to the 動植物檢疫 counter. I don't really know if you need to pick up your cat first or they will pick him/her up for you. You may ask flight attendents. Then~~ 動植物檢疫 personals will follow you to the 檢疫站 either in Taiwan University or the one in Taichung.
Sorry I can't type Chinese for few days. You can search an article I posted in this board.