Hmmm. UPDATE: (sorry for misunderstanding the original question at first)
I just realised... I have that feature turned off for all of my programs except some Wine windows programs. OKAY... so I just remembered that you have to manually select|enable the feature of your panel in your DE (desktop environment).
For example, in XFCE (Desktop Environment) you right-click on the panel and choose "Panel Preferences...". Next select the "Items" tab. Then click "+Add". Then choose from the list of "Add New Items":
Window Menu
-or-
Window Buttons
Either one will give you iconized representations of currently running programs on you panel/taskbar/systray.
It took me a long time to realise that this is what was missing. It has nothing to do with *.desktop files (I did test that too, however).
Sorry that I can't explain how to do it for other DE's such as Cosmic, or Gnome, or LXDE, or whatnot.
However, on my XFCE test system, once I manually added the "Window Buttons", I could see the LMMS logo for selecting the corresponding LMMS program. Also, another LMMS logo was generated for the LMMS project; in other words, it's working.
The XFCE "Window Menu" option is more compact; it's just ONE icon that changes according to whichever program currently running has top focus (is selected and active). You can left-click on that icon and get a dropdown menu of other running programs to choose from, each with an icon and a name.
So ultimately, I am NOT confirming this problem(?) for XFCE desktop environment on (Arch) Linux. I think the way to get the logo representation is through the desktop environment's panel preferences. Again, on XFCE, you could also get there from the terminal: (or Application Finder)
$ xfce4-panel --preferences
Also, after you add the panel items, you can right-click them for "Properties" (more options), or "Remove" to get rid of them. If you're inside of the Preferences Items list, there's a tiny icon of a document with a wrench on top of it; that's the icon to click to edit the properties of a currently selected Panel Item.
Good Luck.
At least for XFCE, I think this is [SOLVED].
Let us know if that worked.
I will look into the settings for Cosmic and maybe some other DE's later.