You have probably seen the pictures before, but the first link works if you copy and paste, and add underscores before and after Species. Here on MO they are removed and the word between them changes to italic letters. I forgot about that..
The virtual Mycota pages are very interesting, but as in any other mushroom site, I don’t think the ID:s are always reliable. At least not in the Calocera cornea case.

After a bit more research I think it’s Calocera fusca.
Your first link doesn’t seem to work but I found this.
http://nzfungi.landcareresearch.co.nz/...

and saw viscid stems below, so you are probably right about Calocera.

I also found this page:
http://virtualmycota.landcareresearch.co.nz/...
There must be some mistake with the ID, none of those look like cornea.
Here’s what they should look like:
http://www.hiddenforest.co.nz/...

I,m not sure either, it looks a lot like other images of C. calocera that I have seen, maybe the surface looks wrong because it’s so dry, the species you mention M. fistulosa is not known from NZ as far as I know.

I guess I could collect a specimen and have a closer look!

It doesn’t look like Calocera to me (they have a viscid, glossy surface),
more like a Clavariadelphus/Macrotyphula, perhaps something around fistulosa.