Well, the first thing we have to establish is that there is no way for the Mac to know " whichever monitor I'm currently looking at". But if you mean which monitor has focus, then yes, that can be done. Most, if not all, auto-display windows will display on the monitor that has focus, meaning the monitor for which the Apple Menu Bar is displayed as active/enabled.
AFAIK, there is no command or API that will set the focus to a specific monitor. To control that, you have to click on a window, or desktop, of the monitor. EDIT: I have found a method using Desktop Spaces that may work.
Please review and test the below Macro to see if it will work for you.
You will need to make changes to the Macro, and maybe to your System Preferences for Mission Control. See Macro Release Notes for details.
The way you use this Macro is to trigger it IF the wrong Monitor currently has focus. The Macro will move the focus to the other Monitor.
I believe (hope) that by changing the Monitor which has focus will solve most, if not all, of your requests. In either case, let us know how it works for you.