Both are listed as ‘atmosphere’.
The first can be part of 空気読めない (KY), so ‘atmosphere’ in a figurative sense. But I thought it also meant plain old ‘air’.
The second one seems to be more like the ‘planetary’ sense of atmosphere?
Which is more common? Are there other ways to understand them?
[1] “Atmosphere” refers to the entire gas surrounding the Earth, and also uses other celestial bodies like “Martian Atmosphere”.
[2] “Air” is the gas that constitutes the lower part of the Earth’s atmosphere. It mainly contains oxygen and nitrogen, and contains a small amount of argon, helium and carbon dioxide. Also, like the “flowing awkward air”, used about the surrounding ambience.
空気 - You’re absolutely right. It means either air, like air you breathe, or atmosphere, like the mood of the environment you’re in. Like 空気を読んで! (Read the air/mood!)
This is the one I have heard regularly during my time in Japan.
大気 - is more environmental and I haven’t heard it very often and never in context of the mood.
So you’re spot on.