Where Does /var/lib/nginx/proxy Come From?
Solution 1:
Turns out the issue was that /var/lib/nginx/proxy is the proxy_temp_path. Normally nginx will download to this temp path and then move the files over to the real cache directory once they've finished downloading.
We must have had a lot of things that didn't finish downloading, or never made it over into the main cache, so the disk filled up.
The solution in this case was to just turn use of the temp_path off (with the use_temp_path=off parameter on the proxy_cache_path directive on nginx 1.7.10 or higher).