First, using the attribute model-type both in the outer and inner directive led to the inner's link function getting executed twice (once for the controller-owned HTML, once for the template HTML), overwriting your formatting. So you need (or at least I think you should) disentagle the attribute for the two directives by using a different attribute for the inner directive:
template: '<div><input type="text" ng-model="ngModel" my-type="modelType" /></div>'
and consequently rename the inner directive to
myApp.directive('myType', function(){
Next, since the inner directive now does no longer get called by the compile step for the outer one, you need to compile the template for the inner directive in your post-link function
This can be done like this:
link: function (scope, element){
$compile(element.innerHtml)(scope)
}
This leads to attrs being always the same, so you can't use it to test for your testType:
$$element: U[1]
$$observers: Object
$attr: Object
myType: "modelType" <-- see here
ngModel: "ngModel"
type: "text"
Therefore you need to to look for the actual value inside scope:
if(scope.modelType == "testType") {
Lastly (and frankly I would be happy if someone could explain this to me, I don't understand it), I had to define the modelType scope property in the outer directive as one-way bound via
modelType: '@modelType'
I put this together in an updated fiddle here: http://jsfiddle.net/nkop2uq0/8/
NB: I am far from understanding the intricacies of Angular's directives. As much as I like the framework, the number of ways things can be achieved is mind-boggling. So you should try to improve my answer, it is most likely not best practice. And drop me a line if you find bad ideas in there...