Desktop screenshots: Clicking button captures partial screenshot

Wires up the button in the region box to the business logic to request
taking a partial region screenshot. Also renames the button component to
not be specific to screenshots.

Bug: 417534202
Test: Manual
Flag: com.android.systemui.desktop_screen_capture
Change-Id: Ia732d9cfb82015c463e90f3e9d5175a3e579a48c