at 7:41 am or so on May 1st 2024 started this test with debug on minimal and attached the resulting zip file to this post (dual monitor set up, bottom is main external hdmi monitor #2, top is secondary hdmi #3, (internal laptop display #1 disabled)
First tried Mozila Thunderbird on #3 shows titlebar button in windowed or maximized mode. sending window to #2 works but titlebar button only shows in windowed mode, not maximized mode. sending back to #3 from windowed mode works, and titlebar button shows in both modes again. repeat sending back to #2, same issue button only shows in windowed mode.
Then tried test with the default Windows Media Player app starting with #3. Titlebar button shows in windowed mode only, does not show in maximized mode (so is different than Thunderbird). send to #2 while in windowed mode works, same issue, the button only shows in windowed mode. send back to #3, same issue.
Stopped testing at this point, but also have the issue with SichboPVR4, but not quite sure how to describe because the app never uses a normal Windows titlebar, it is an overlay with transparent background in windowed mode, all icons dissapear after a moment except your titlebar button, which stays. not sure if that app maximize button actually maximizes the window or enters fullscreen mode, but in any case your button now is missing in that mode whereas previously it remained like in windowed mode (after the Sichbo icons disappear).

Enabling that setting resolved the issue for Thunderbird and Media Player (Thank You, I should have looked for that myself even though the help page describes it's purpose a little differently), but not SichboPVR4 (button only appears in windowed mode on either monitor). Posting a new minimal log result with a test of SichbopVR4 at 8:06am.

Well my success was short lived, after leaving for a couple hours and logging back in, the issue has returned for Thunderbird and Media Player. The only change I made after the success was moving the titlebar 10 pixels left and 5 down, then I rebooted and tested the apps and everything was still good at that point. I checked and the setting you had me enable is still enabled.

a new version of the issue happening now. not sure if this is the only app combo that does it, but this combo repeatedly does it.
If I have the native PhotosApp viewer and the Chrome browser open in the same monitor with the photos app on top, the photos app titlebar button actually moves the chrome browser window instead. Turned on the titlebar tooltip debug feature and confirmed that the Photos app titlebar button is calling the Chrome browser app. Trying various restore/maximize arrangements of both apps does make a Photos titlebar button appear that points to the Photos app, but the problem always reappears in the same arrangements.
Occasionally right after rearrangement the Photos app will have two titlebar buttons for a couple seconds, a good one and a wrong one, in different horizontal positions. Eventually both buttons disappear if the Photos app is maximized.
Since I have noticed that different kinds of apps treat the horizontal placement of the titlebar button differently, I turned off the Use Aero Titlebar Positioning and restarted, but nothing changed. so i tried switching titlebar mode to Classic instead of Aero and restarted your app again. buttons look different but issue remains, including how some apps put your button more to the left than others.

another observation: If I have the Prime Video app maximized, then shift focus to an app like Chrome browser to make it appear on top of the video player, I cannot see your titlebar button in the browser if it is maximized. I do see it if I restore the browser down to a window.

latest beta is actually v11.0, Beta 5
installed it and tested every scenario I put in this thread.
seems to be working properly in every case, except the occasional presence of correct titlebar button AND the button for the hidden window beneath it persisting for a moment on the top window. The wrong one does disappear after 1-4 seconds or so.
Overall it does seem that the behavior of the app is different, there is more "blinkiness" of windows as I move them between monitors, like your app is refreshing something differently. But it is ok.
A different problem that I never mentioned before because I don't know if it is your app or not, but I will hope it is resolved too: On occasion when I boot up or log in one or more open windows have shrunk down to a very tiny pixel size, so small it is hard to find them even if I activate them. On a rare occasion they have been moved offscreen (meaning off all screens even though open) If this issue is known to you let me know, and if it occurs again I will let you know.
Thanks!
P.S. forgot to say that even though entire windows blink differently now when moving monitors, the actual titlebar buttons do shift horizontally like they were when changing from fuul to windowed. Ans apps like Excel used to have the button shift horizontally while saving the file, but then go back to the normal position (I may have macros doing things during a save that change the window mode but don't update the screen so I dont see it, but your app does).