Edge executable file location in registry

Sorry, I am not a 100% sure, although that "House Icon" itself is most likely from at least either ieframe.dll (icon-1) or even iexplorer.exe (icon-9). One vaguely related entry with the iexplorer.exe (icon-9), is in this key :

Computer\HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\IEXPLORE.EXE\DefaultIcon
C:\Program Files\Internet Explorer\iexplore.exe,-9

However, I am unable to test that "House Icon" in any Right-Click Menu as I still haven't set up that .URL Registry change.

Update:

The icon change in Computer\HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\IEXPLORE.EXE\DefaultIcon did not work.

( I was able to add to the .URL "Open With" context menu by going to Settings > Choose Default Applications by File Type > .URL > = change to Internet Explorer. This then produced the choice of Edge or Internet Explorer in the context menu. It is important to note that by doing it through the above settings, the "Hash" string under the "UserChoice" key is validated by Windows - otherwise if I copied your Hash string - it got deleted as soon as I Right-Clicked on a .URL file and the choice also did not appear ).

The Notebook I'm on now has the Edge at "C:\Program Files (x86)\Microsoft\Edge\Application\msedge.exe" although in Edge, Help and about describes it as 64-bit. I find no .exe in Program Files or ProgramData folders.