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Does use of the terms "astronaut" and "cosmonaut" violate neutral-point-of-view policy or constitute systemic bias? (No.)
No. Astronaut was used by the United States for its first space travelers in 1961, as was cosmonaut by the Soviet Union. In today's age of the International Space Station, the terms serve a valid distinction, as persons launched by the US are called astronauts and persons launched by Russia are called cosmonauts, regardless of their nationality.
Should the term "taikonaut" be used to designate Chinese space travelers? (Not necessarily.)
No real-world consensus seems to exist. Taikonaut seems to be a neologism favored by some English-language news media organizations. The Chinese government officially uses both astronaut and cosmonaut in text issued in English and Russian, respectively. The native Chinese phrases used officially for "space navigating personnel" do not translate easily into a single English word. "Taikonaut" is an Anglicization of the Chinese for "spaceman" popularized in Hong Kong and Taiwan.