I am running 22.04.1 LTS and installed both the Ubuntu system 1.8.19 arduino ide and also have the direct download (.AppImage) of the 2.0.0 arduino ide located in my ~/bin. I successfully added a custom .desktop file to my ~/.local/share/applications as online resources describe (from copying and modifying the system installed version for the app). The result is that I now have a launcher which calls my intended 2.0.0 ide and I am happy with that. However, there is an issue in that there are no little green dots next to the launcher icon (the ones that show up on the left hand side per window open). I will attach a screen shot with multiple windows open.
The original .desktop is as follows:
[Desktop Entry]
Name=Arduino IDE
Comment=Open-source electronics prototyping platform
GenericName=Arduino IDE
Exec=/usr/bin/arduino %f
Icon=arduino
Type=Application
Terminal=false
Categories=Development;Engineering;Electronics;IDE;
MimeType=text/x-arduino
Keywords=embedded electronics;electronics;avr;microcontroller;
StartupWMClass=processing-app-Base
I modified the .desktop according to this thread -- Main menu entries created with Alacarte - No icons in dock. Here it is:
[Desktop Entry]
Name=Arduino 2.0 IDE
Comment=Open-source electronics prototyping platform
GenericName=Arduino 2.0 IDE
Exec=/home/tniles/bin/arduino %f
Icon=arduino
Type=Application
Terminal=false
Categories=Development;Engineering;Electronics;IDE;
MimeType=text/x-arduino
Keywords=embedded electronics;electronics;avr;microcontroller;
StartupWMClass=arduino ide, Arduino IDE
And, yes... I have tried logging in and out and rebooting. :-) So any ideas or help would be appreciated! Thank you!
Additional resources: https://help.ubuntu.com/community/UnityLaunchersAndDesktopFiles
Photo with system install ("working", green dots visible):

Photo with custom .desktop and 2.0 install (not "working", green dots invisible):

Update: in generating these screenshots, I now realize there is something else going on, in that the generic gearbox icon is the one for the custom .desktop. However, it's tooltip says "Arduino IDE" which matches the system .desktop, even though the Arduino icon's tooltip says "Arduino 2.0 IDE" which matches the custom setup. The plot thickens...