Hi Ben,
There aren't actually cliffs you can fall off like Mt Anne, unless you go looking for them. There are steep sections of rock, where you need to be careful, especially when it's wet. At times you should take extra care on the track-y bits as well, as it slopes steeply away in places. But I can't recall any precipitous bits, and my radar is reasonably good for that sort of thing.
I've come down on wet rock, without significant issues. But we took it slowly, and probably crouched down on some steeper bits.

- The rock isn't quite as vertical as it looks here, and there is no cliff to fall off if you slip a bit.
The summit itself is a gentle dome (at least the way the track brings you). No exposure, unless you seek it out away from the high point. You can find big cliffs if you want to.
The main obstacle people talk about is an scramble up and down the rock where there is an awkward rock above you, getting in the way.

- The awkward rock scramble
Again, no cliffs there. We were definitely careful coming down that bit in the rain. Parks probably say that because not everybody is. And so much depends on the individual's capabilities.
Can you give us an idea of a walk/climb you've decided not to continue on, because of the vertigo?