Here is what I found comparing jwm, icewm, LXDE2, and fluxbox.
Icewm, fluxbox, and LXDE2 all recognize and use the app's internal icon spec for the titlebar icon. None of them fall back to a /usr/local/share/pixmaps icon, but LXDE2 does use the Icon= field in the .desktop files to make a menu icon. So in LXDE2 a menu icon is there as long as Icon= is specified and valid in the .desktop file. So you can have a generic title bar icon but a specified menu icon in LXDE2.
Jwm does not use the app's internal icon spec, and instead relies totally on a /usr/local/share/pixmaps icon that is of the same name as the executable. And it does not appear to do that reliably, since links2.scm and frozen-bubble.scm do not show a title bar icon at all no matter how the files are named, though both of those apps by default show a title bar icon in the other WM's and DE. CORRECTION: Jwm does seem to obey the icon spec in some apps when there is no pixmaps icon, but there is still some hit or miss.
So my conclusion is that the effort is not worth the result to further pursue renaming and restructuring extensions to only benefit jwm. It would be different if it was also fixing a same problem in all the freedesktop DE's. But they don't have that problem, they obey the app's standard icon spec. Also, it is not just our build as I tested Debian's jwm and saw the exact same behavior.