The problem is that the MVC pattern was designed in a system that doesn't really exist anymore. It was invented in Smalltalk at a time when UI libraries did not exist. To make a window dialog you drew all the boxes, highlighted the appropriate squares, made sure that the text you were drawing ended up in the right spot...etc...
Imagine what it would be like to write a dialog app using nothing but one large canvas. That's the world the MVC comes from.
A "view" in this system was a text box and it was a class that was responsible for drawing the box, the text, drawing selected areas, responding to changes in the text, etc...
A "controller" was another class that took mouse events that occured within this box like mouse moving, key down, key up, clicks, etc...and it would decide what happened. Should we change the text? Should we change the selection? Stuff like that.
A "model" was yet another class that represented the basic data and state of the component. A text box model would have the text of course, the font, selection, etc...
As you can see, in a situation like this the three components are very entangled in the representation of a single idea. It makes sense in this context to speak of a "triad".
Today, if you're working on creating a UI library and using raw drawing commands you might do something similar. But the application of the "MVC" pattern has spread beyond its initial purpose. Now days you have a "view" that may actually be a complete dialog, and a controller that's responding to events like "textChanged" or "buttonClicked". The model in today's MVC is normally something fairly disconnected from the system (but generally linked to the view by providing an observer interface of some sort) and there may be many views associated with the one model.
In a system I recently architected for example we had around 10+ views all observing a single document "holder" and its active document. A main drawing interface interacted with the layout of the document, various property views that observed the selected item and provided a record interface, and a smaller scale representation of the main view that showed the entire document instead of just the visible window. Some of these views had controllers of varying complexity that turned GUI events into changes to the document, which would in turn notify its various views.
Can you still call such a relationship a "triad"? Perhaps, but I think it implies too much of the former, older application of MVC.
Could you share controllers with different views? Depends on how similar the views are. I've found that generally speaking this type of object has behavior to specific to the view it's controlling AND the model it is manipulating to be very reusable...but there's always exceptions.
It depends on the type of MVC you want. I suggest you try the Michael Feather's "humble dialog box", which is also called "model view presenter (MVP)", and probably what you are looking for.
In this variant, the view has a refererence to the controller, and the controller has a reference to an abstract interface of the view. So the view can provide methods for registering listeners, those methods are also available in the interface, and the controller will call these methods without having direct access to the UI.
Best Answer
The controller controls the flow of activity. The user performs this action, the controller passes the view data to the domain which does whatever it needs to do then, based on the response(s), the controller tells the framework which view to show next (and gives it enough data to do so).
The controller must thus be coupled to the domain model, to some extent. ie. You could put a service layer in between but, by strict definition, that becomes part of the domain.
It is also coupled to the view data but not the view itself. ie. it simply says "show the customer view using this customer detail." The framework then decides where it should find that view.
Now this should allow you to decouple the domain model from the view, by using a view model of the same data. Some developers do this, some don't, and I do think it's largely a matter of personal preference.
In Rails, you are very much encouraged to push the domain objects (ActiveRecord) to the view and trust that the view doesn't take advantage of that access (eg. you shouldn't call customer.save from the view, even though it would be available).
In the .NET world, we tend to reduce risk by not allowing things that shouldn't happen and, possibly for that reason, it seems to me that the detached view model is more popular.