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Perhaps you misunderstood, it is not about icon you see in the tray (left from clock/date), but it is about an icon which is displayed on in Notification Area Icons - you get there by click on Customize (Taskbar and Start Menu Properties). This is not related to an icon in the tray (it goes away when I close Electrum) or about some Electrum process that is still working in the background.
I was indeed mistaken, sorry for the confusion. I see that your concern is now mine as well, my menu also shows my balance, even when Electrum isn't running. I don't share my computers, nor send them out for repair, but it's still something that should be addressed in my opinion.
just enough to be "dangerous"
Apparently, lol. That's a great hack, you should submit a pull request to have that feature added to the next release.
Are you using Visual Basic to modify the code and build binaries?
HCP, so you played around with the code and you add this option to Electrum? I am asking this because I check settings in 3.3.7 and I do not see that option. I still haven't updated to 3.3.8, it doesn't really matter to me since I only use Electrum in combination with Ledger.
Like TryNinja said, those are features that HCP added himself. That checkbox is not a feature in version 3.3.8.