[iOS][BWG] Setup and check BWG background active state for animating UI

Changed to having a UIShown bool and activeInBackground bool in the BWG
tab helper to keep track of whether the BWG session is active for a
given WebState in the background. Use this new information both for
tab persistence but also for the animation bool when showing BWG UI.

Bug: 429250847
Change-Id: I83f8b9f46bb2d7cf2f841595cd1cef5934142764
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6704111
Reviewed-by: Adam Arcaro <adamta@google.com>
Commit-Queue: Nicolas MacBeth <nicolasmacbeth@google.com>
Cr-Commit-Position: refs/heads/main@{#1482373}