Just to emphasize because i first i had trouble understanding this as well.

It's the sum of the initiative from ALL your active ninjas that count. So the initiative of your move 1,2,3 AND 4 ninja added up together.

The team that has the higher initiative overall gets to keep their barrier.

The position of your ninja or initiative of the barrier ninja wont get you the barrier if you loose it to anothet team. You need to increase the initiative af your lineup.