[IMPROVEMENT] Allow hiding the tray icon while keeping UniGetUI running in the background

### Please confirm these before moving forward.

- [x] I have searched for my feature proposal and have not found a work-in-progress/duplicate/resolved/discarded issue.
- [x] This improvement refers to an existing feature.
- [x] This improvement is not a bug.

### Describe the improvement

Please separate system tray icon visibility from the behavior when closing the main window.

On Windows, I wanted to hide the tray icon while keeping UniGetUI running in the background. After disabling the existing Close UniGetUI to the system tray setting and closing the window, the application appears to exit instead.

Suggested independent settings:

- Keep UniGetUI running in the background when the window is closed.
- Show system tray icon.

With background operation enabled and the icon disabled, closing the window should leave background update checks running without a tray icon. Ideally, notifications should remain separately configurable where technically supported. If there are limitations, please explain them next to the setting.

Reopening UniGetUI from its normal shortcut should restore the existing window, and an explicit Quit command should remain accessible in the main UI. Existing defaults can remain unchanged.

This request is about hiding the icon entirely, not changing its theme or artwork (as in #4374 and #4838), nor merely moving it into Windows tray overflow.

### Describe how this improvement could help users

Users could keep background update checking while reducing tray clutter. Separate controls would also make the settings clearer: window-close behavior and icon visibility are different preferences, but the current wording makes it difficult to discover whether they can be configured independently.

Start by tracing the existing “Close UniGetUI to the system tray” setting and the window-close, tray, background-update, notification, reopen, and Quit flows. Done means background checks can continue without a visible tray icon, reopening restores the window, explicit Quit remains available, and existing defaults are unchanged.

Written by the indexing model from the issue text.