New South Wales
The bit that confuses me is if you start your turn in a space with a monster and use the move action to leave it. That seems to contradict these two rules.
I'm voluntarily leaving the space, but doing so using the move action which doesn't trigger an evade check.
Unless the clarification is that you don't have to perform two evade checks - 1 for taking the move action, and 1 for leaving the space?
You have the original timing wrong, which makes your last supposition correct. You have to do the Evade check before doing an action which isn't Move nor Attack. You also have to do it whenever attempting to move out of your space voluntarily.
So given that you can only leave the space after you successfully triggered the Move action, you would have to Evade twice if the Evade check rules didn't exclude Move actions; once before the Move and once during the Move when you actually leave the space.
The way it's worded can be confusing, but it is necessary.