Well,
Every angular directive create new scope
In the code below
<div ng-if="$state.current.url!='/splash'" >
<button ng-click='x = ! x'>toggle</button>
</div>
<div ng-include="'pages/include/search.html'" ng-show='x'></div>
The ng-if directive create new scope.And withing this scope
the value of x is updated on button click.But this new value of x is not accessible outside this ng-if div as it is local to that scope and it is primitive type.
Since this x is primitive type so there is no data update as reference are differant.
You should use object model instead.
Here is the updated HTML
<div ng-if="$state.current.url!='/splash'" >
<button ng-click='x.showHide = ! x.showHide'>toggle</button>
</div>
<div ng-include="'pages/include/search.html'" ng-show='x.showHide'>This is div</div>
define x like this one in your controller.
$scope.x = {showHide:false}
EDIT-
In your first woking HTML, there is not directive on div.So, both these DIV come under same scope.So,x accessible across this two DIV with updated value.
<div>
<button ng-click='x = ! x'>toggle</button>
</div>
<div ng-include="'pages/include/search.html'" ng-show='x'></div>