Ok, it turned out that something between my computer and the head server was filtering POST requests with wp-admin and post_type[s] anywhere on the request.
Solution: move to https
Thank you!
So I started from a working configuration without the proxy, after introducing the proxy, things started to break.
I also tried with Apache as the internal web server and the result was the same.
What I find very weird is that some POST requests do go through, I can write post and pages, and even the admin-ajax.php watchdog returns without problems. Only in those specific tasks, the request seems to be lost in limbo.
