No, there is no clash. I did't expect you to read it like that. Even though, i could have written that, there are languages i write much better in.
I tried to lead you in finding out the things by yourself.
The default controls do what you need: A double jump when in the air.
If you change one of the default controls, you need to take control of them all. And suddenly, as by magic, you will see the solution just by yourself.
If i understand you correct, you want a double jump when in the air only if 'z' is pressed. In that case you also need to alter the 'up arrow', so it dont double jump. Easy as that, when you take control.
Not that i would like play this way. I still prefer default to play.