heading to Taiwan

Hello from Korea,
I am planning my trip to Shanghai and Taipei with my three friends (all Koreans)
My flight plan is
Feb 20 / Seoul(GMP) - Shanghai(SHA)
Feb 26 / Shanghai(PVG) - Taipei(TPE)
March 3 / Taipei(TSA) - Shanghai(SHA)
Shanghai(SHA) - Seoul(GMP)
What I'm curious about is am I allowed for 144-Hour Visa-Free Transit at Shanghai? one of my friend told me that Taiwan does not count as third country but here it is said that it DOES count in term of immigration affairs. I just want to make sure before it's too late.