No playback options when right clicking play button Logic Pro X

It all depends on how you have your mouse programmed. If you are using an Apple mouse, usually the right click sends a control click. Which brings up a contextual menu in most programs. Including Logic. If you wish to make your right click behave the same as a left click, you'll need to change that settings in the Mac OS mouse settings. Not in Logic. Or you might need to your a 3rd party preference. Such as Logitech to change it.

I see there is a setting in Logic under General>Editing for the right mouse click, but it seems to be only for editing, not other general things. Anyway, glad you got it working the way you wanted.