The version I have been using since the beginning is in camel case, so “EarthOS”. As a single word, it has the advantage of being more unique in Internet search, compared to just the two words that have a meaning of their own. But that’s not much of an important consideration.
For naming a team / organizational unit, I like both versions actually. “Earth OS” is more descriptive, while “EarthOS” appears more like an identifying name of a single product / system. The descriptive one “Earth OS” might make more sense to express that “it’s a unit that cares about sustainable civilization” rather than “it’s the unit that develops EarthOS, you know, this crazy publication series”.
In any case, EarthOS / Earth OS is good as the title to put into the edgeryders.eu main menu etc., but the long form when talking to people would have to be something like “I’m part of the Earth OS division / team / unit of Edgeryders”. “I’m part of Earth OS” makes rather no sense: the unit cares about developing / maintaining / contributing to the Earth’s Operating System, but is not identical with it.