I am not sure why you were creating some many copies of the service I have simplified your code and I think it should work as you expect:
js
var testModule = angular.module('testmodule', []);
testModule
   .controller('QuestionsStatusController1',
    ['$rootScope', '$scope', 'myservice',
    function ($rootScope, $scope, myservice) {
       $scope.myservice = myservice;   
        $scope.myservice.confirming = "asdsadasdd";
        $scope.ForceBinding = function(){
        $scope.myservice.newValue = $scope.NotBinding;
    };
    $scope.$watch("NotBinding",
        function( newValue, oldValue ) {
          $scope.myservice.newValue = newValue;
        }
    );
    $scope.$watch("confirming",
        function( newValue, oldValue ) {
          $scope.myservice.confirming = newValue;
        }
    );
    }]);
testModule
   .controller('QuestionsStatusController2',
    ['$rootScope', '$scope', 'myservice',
    function ($rootScope, $scope, myservice) {
      $scope.myservice = myservice;
    }]);
testModule
    .service('myservice', function() {
      this.xxx = "yyy";
    });
and html:
<!DOCTYPE html>
<html ng-app="testmodule">
  <head>
    <script data-require="[email protected]" data-semver="1.1.5" src="https://ajax.googleapis.com/ajax/libs/angularjs/1.1.5/angular.js"></script>
    <link rel="stylesheet" href="style.css" />
    <script src="script.js"></script>
  </head>
  <body>
    <div data-ng-controller="QuestionsStatusController1">
     1<br> <input ng-model="myservice.xxx" />{{myService.xxx}}<br>
      <hr>
     2<br> Dynamic Binding :       <input ng-model="NotBinding"/><br>
      Click to dynamic Bind <input type="button" value="bindNow" ng-click="ForceBinding()" >
    <hr>
    3<br>Binding directly to the service model<input ng-model="myservice.confirming">
    </div>
    <hr><hr><hr><hr><hr>
    <div data-ng-controller="QuestionsStatusController2">
   The value of xxx is: {{ myservice.xxx }}
   <hr>
   Dynamic Binding Problem Value  : {{myservice.newValue}}
   <hr>
   Direct binding to the service object  : {{myservice.confirming}}
    </div>
  </body>
</html>
<!DOCTYPE html>
<html ng-app="testmodule">
  <head>
    <script data-require="[email protected]" data-semver="1.1.5" src="https://ajax.googleapis.com/ajax/libs/angularjs/1.1.5/angular.js"></script>
    <link rel="stylesheet" href="style.css" />
    <script src="script.js"></script>
  </head>
  <body>
    <div data-ng-controller="QuestionsStatusController1">
     1<br> <input ng-model="myservice.xxx" />{{myService.xxx}}<br>
      <hr>
     2<br> Dynamic Binding :       <input ng-model="NotBinding"/><br>
      Click to dynamic Bind <input type="button" value="bindNow" ng-click="ForceBinding()" >
    <hr>
    3<br>Binding directly to the service model<input ng-model="confirming">
    </div>
    <hr><hr><hr><hr><hr>
    <div data-ng-controller="QuestionsStatusController2">
   The value of xxx is: {{ myservice.xxx }}
   <hr>
   Dynamic Binding Problem Value  : {{myservice.newValue}}
   <hr>
   Direct binding to the service object  : {{myservice.confirming}}
    </div>
  </body>
</html>
Check the differences between your code and mine and you'll find what was wrong, for instance the div for the controller 1 was closed before number 3 (that was why it wasn't being updated).
You can see it in this plunkr.