I just found out today that exactly the same behaviour was happening with me (web contacts -> android : OK; android -> web contacts : FAIL). I've always worked without any problem (the sync used to be instantaneous).
Today I found out that sometime in the past, I've had switched to the web...
Thanks Paul!
That's why even with a Nexus, I keep using CM. Tweaks, Tweaks and.... more Tweaks available.
Just applied your sugestion with ProxyDroid (which although I want its visual notification, I don't need the LED one).