Apple - Keynote Play presentation on 1 display and Just see my desktop on my second

I had a similar problem and found a surprisingly simple workaround on the Apple Support Forum (https://discussions.apple.com/thread/5522746?tstart=0)

I needed a way to show my presentation on a webinar and still have access to the webinar control panel so I can see questions when they come in without having to close the presentation first.

The SOLUTION: You basically just need to export your completed Keynote presentation as an HTML file and open it in a browser. Then simply make it full screen on the screen or projector you're showing it on.

I used Chrome and found that the address bar still appeared in full screen view. To fix this I simple unchecked the "Always show toolbar in full screen" option in Chrome's View menu.

Solved the problem for me. Transitions worked too.

I know this thread is pretty old but I hope that helps someone searching for help with now (like I was)

This issue has been bugging me for ages. However, Apple has finally updated Keynote, half-assed but still.

In Keynote 10.1, there is finally the Play > Play Slideshow in Window option which does exactly that.

You'll have a window (where still the window bar shows up when you move the mouse, well done Apple!) with your presentation, you can resize it and move it wherever you want, share it with Zoom or another video conference tool and still be able to see your Desktop.

After tons of research around the web, I found a way to run other applications while running a Keynote presentation. Within Keynote's preferences, be sure the box for "Allow Expose, Dashboard, and others to use the screen." Next, set up Mission Control (formerly known as Spaces) do have your Keynote window on Desktop-1 and your other application's window on Desktop-2. Now hit "Play". At any point during the presentation, press "F" (to pause the slide show); Command-2 (to switch to your alternate application); Command-1 (to switch back to Keynote); and Space Bar (to resume). Your clients will see only one slide for the duration, and will not see anything you are doing on virtual Desktop-2.