144 Hour free transit: Transit definition

A transit by definition looks back only at the inbound flight vs outbound flight, correct?

If I bought a RT ticket to Taiwan from the US, but now want to add a short trip to visit Shanghai in-between my stay in Taiwan. I want to fly US - TW - China - TW - US. Am I correct in that this would not qualify me for the 144 hour free transit?

So, would I need to cancel my return leg from taiwan and re-arrange my return to look like TW - China - US to qualify for the 144 hour free transit?

Finally, would an email of a confirmed ticket from an online agent qualify as proof?

Thanks.