The fact that the Standard rendering does render these inners, but as you wrote Humanitarian / Cycle Map / Transport not, does make it likely this is an issue of older versions of osm2pgsql being used.
Contacting the developers is probably only really sensible if you know the exact versions of the software being used by the renderings that fail.
I would definitely not change your tagging / mapping practice to suite a specific renderer or (older) version of an import tool, unless the MP are simply broken and invalid in JOSM, in which case you need to fix them. That doesn't seem likely here though, considering they do render fine in Standard style.
The **The solution in your case seems to upgrade your own software stack.stack.**