AppImage: leave the menu entry to launchers that set DESKTOPINTEGRATION (#1635)

Thanks @lasselian.

packaging/linux/AppRun

@@ -11,7 +11,9 @@
 # the launch untouched. Opt out with SOLIDCRAFT_NO_DESKTOP_INTEGRATION=1 (the pre-rename
 # SOLVECRAFT_NO_DESKTOP_INTEGRATION still works: compatibility shim, docs/compatibility.md),
 # or for good with an empty file $XDG_DATA_HOME/solidcraft/no-desktop-integration. Nothing is
-# installed either when appimaged (go-appimage) already made an entry for this AppImage.
+# installed either when appimaged (go-appimage) already made an entry for this AppImage, or when
+# a launcher that manages AppImages (Gear Lever, AppImageLauncher) started it: it sets
+# DESKTOPINTEGRATION and owns the menu entry.
 
 set -u
 
@@ -42,10 +44,21 @@ integrate() {
   src_desktop="$APPDIR/usr/share/applications/$DESK_ID.desktop"
   [ -f "$src_desktop" ] || return 0
 
+  apps="$data/applications"
+  # Gear Lever and AppImageLauncher write their own entry and start the AppImage with
+  # DESKTOPINTEGRATION set; treat that like appimaged: no entry from here, and drop the one
+  # this script wrote earlier (it would otherwise come back after the launcher's is removed).
+  if [ -n "${DESKTOPINTEGRATION:-}" ]; then
+    mine="$apps/$DESK_ID.desktop"
+    if [ -f "$mine" ] && grep -q '^Exec="' "$mine"; then
+      rm -f "$mine"
+      command -v update-desktop-database >/dev/null 2>&1 && update-desktop-database "$apps" >/dev/null 2>&1
+    fi
+    return 0
+  fi
   # appimaged integrates AppImages itself (appimagekit_<hash>-<name>.desktop, Exec= this file):
   # a second entry from here would show the app twice, so drop an entry this script wrote earlier
   # (only one it wrote: it points Exec at a path in quotes) and leave the rest to appimaged.
-  apps="$data/applications"
   for other in "$apps"/appimagekit_*.desktop; do
     if ! { [ -f "$other" ] && grep -qF "$appimage" "$other" 2>/dev/null; }; then
       continue