[PiP2] Fix Btn-nav and direct enter PiP from Split

Make sure DefaultMixedHandler properly handles split-to-pip
with PiP2 flag on when going home from split or requesting a direct
enter PiP from one of the stage tasks (can be trigered via back gesture
in certain apps like YT). This needs to be properly followed by a
DISMISS_SPLIT transition started by StageCoordinator in the
single-activity PiP case.

Issues that are yet to be fixed:
 - back gesture does not trigger an alpha enter PiP animation, and
instead has PiP window appear with a jumpcut
 - entering split-to-PiP from a multi-activity PiP app in either stage
is currently not polished

This change also move the AnimationType to PipTransitionController and
allows animation type overrides for both pip1 and pip2.

Bug: 381374430
Flag: com.android.wm.shell.enable_pip2
Test: manually repro the steps in the bug
Test: create a split with a PiP task and tap home btn
Change-Id: Ia867afd77aa08f6d75db1979b3f5ee14eac0f3d8