The application remembers the bounds and specifies where to open the window, so this is WAI. If you move the window to external display and close it, then the window will be opened in the external display next time you start it.

I don't think this makes sense. It may make sense relative to the current screen, but if I click on the toolbar on one screen vs another, there is very clear user intent that the app should open in the given window (by choosing which window in which to click the app icon).

As I explained, it is the application that specify the bounds, not chromeos.
Application may open multiple windows in a way that they're aligned, and we can't randomly modify the bounds.

Fine, but bounds != window. Can't you switch which window it is opened on, based on the window where I clicked the application tray?

We can't ignore the bounds as we do not know the intent of the application.

Another important fact is that application is simply asked to start asynchronously, and application may not open a window, or may open at quite different timing. It may open after you click something on the other display. 

We do open the browser window on the same display as shelf you clicked, and
that's because we control the bounds and we know when it opens.

"We do open the browser window on the same display as shelf you clicked, and
that's because we control the bounds and we know when it opens."

Only if you right-click and select "New window". Surprisingly, clicking on the browser button in the other monitor (where no browser windows are open) minimizes the browser window on the first monitor where a browser window exists (which is likewise very surprising behavior here).

"We can't ignore the bounds as we do not know the intent of the application."

If it's up to the application.... is there a way for applications to identify the monitor on which they were open? And do we have guidelines / help for applications to implement this correctly? Lastly, I encountered this issue specifically while opening Chrome Remote Desktop (an application under our control)... at the very least, could we ensure that this behaves correctly?