That's not at all what i said. I said the tray icon is always there when the tool is open (background mode or not). If you quit the tool, the tray icon goes away.
Whether clicking the X at top right of the window quits out of f95checker entirely (including removing the tray icon), or instead goes into background mode (closes the window but keeps the tray icon) is an option: the one
WhiteVanDaycare showed you.
This option is intended for people who always want f95checker in background and don't want to click the "BG Mode: Switch" button, they can just close the windows and f95checker remains open in background mode.
This setting (and no other setting) has no impact on whether the tray icon is present while the window is open, it's always there, and that's what i thought you were saying wasn't the case.