Astronaut is the original English word. Cosmonaut is a transliteration from the Russian word for astronaut. So if we're speaking English, we should use astronaut regardless of the country they're from. Otherwise we're forced to invent silly new words that nobody understands like taikonaut and what will we call Indian astronauts or whoever's next?

https://en.wikipedia.org/wiki/Astronaut#Terminology

Well, then don't navigate the astres, the cosmos or the stars. They only circumnavigate Earth and a few went just so far as Earth's Moon.

Let's call them Earth orbitonauts. :-)