Javascript works on references. Your array items in app is same reference as items of strategyB items initially with the below statement and when you update the StrategyB items automatically items in your view gets updated(since same reference).
$scope.items = Storage.getItems();
So, when you switch strategy you are not changing the reference of items. It still points to StrategyB items reference.
You have to use the below mechanism to change the reference.
Then you can do something where you can communicate between controllers to change the items reference.
Please find the plunkr I have updated.
$rootScope.$broadcast("updateStrategy");
And then update your item list and others.
$scope.$on("updateStrategy",function(){
$scope.name = Storage.getName();
$scope.items = Storage.getItems(); //Here changing the reference.
//Anything else to update
});