Record mix with DDJ-RZ

I don't remember if there's a way to toggle it with the DDJ-RZ itself, but there is a record button in Rekordbox DJ. Look at the top left of your screen and you should see a bunch of buttons. One of them says "REC" and if you click it, a record button with some level meters will pop up halfway down your screen.

Go into your settings and change the record folder.

I just made a folder DDJRZ, and set the recoding to there.

It worked just fine after that.  I had the same problem you're 

having until I created and changed the folder.