Thinking that a countries independence is determined by who they pay taxes to is ridiculous. The UN recognition makes most sense- the UN is internationally recognised and basically defines independent. Also, I do not think Taiwan counts as an independent country- rather they are like Palestine is a state between independent and not. Also, Armenia, Turkey and People's China all have recognition from the large countries so they are independent.
Your argument is rather circular here. The UN recognition makes the most sense, since everyone recognises the UN, so the UN recognition is the most credible. There is not unanimous UN recognition on Armenia nor Chinas. But like it or not, Armenia is not part of Azerbaijan, Mongolia is not part of Taiwan, and Chinese Taipei does not exist. It's more important to draw maps how they are rather than how we want them to be.
