You can change the name of the Local Area Connection by using Start / Settings / Network and Dial-up Connections, right-clicking Local Area Connection, and pressing Rename.

You can change whether the connection icon is present in the Taskbar by double-clicking Local Area Connection, pressing Properties, and checking the Show icon in taskbar when connected box.

To configure these setting via the registry, navigate to:

HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Network\\{4D36E972-E325-11CE-BFC1-08002BE10318\}\\{GUID of Adapter\}\Connection

To rename Local Area Connection, change the data string of the Name value name.

To change the status of the Taskbar icon, set the ShowIcon value name, a REG_DWORD data type, to:

0 - The icon is NOT displayed when connected.

1 - The icon is displayed when connected.

These settings will become effective on the next restart.