I think you are looking at this in completely the wrong way. A GUI app and a web page are worlds apart so the exact same definition of MVC will never work for both. MVC is more about the ideal: separating certain parts of the app like display and logic.
In PHP (or the web in general), a View is the web page itself: the HTML output. It's not "live" as per your definition, but you simply click links to go back to the controller (i.e. another page request).
The Controller and Model is where things do differ, like you explained. In PHP the model tends to be the data layer, interacting with the database and so on. But it is still modelling the situation, and the controller still controls the application flow, if only once per page load.
So the name "Model-View-Controller" is perfectly logical, albeit a different implementation in GUI apps vs web apps.
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.
Best Answer
Model: Fields that belong to the object, methods that help to get/set data from the object (a fullname accessor that returns first + last name)
Service: Methods to perform operations with one or more models, see 'unit of work', transactions, etc...
Employee::create should just take a set of data, perform model validation if necessary, and return an Employee Object.
EmployeeService::hireEmployee might create the employee, send them a welcome email, create a mailbox, make them a sandwich, etc... it may return the set of data, or a result code, etc...
This can also affect validation:
Model Validation: Employee must have a id, first and last name, and birthday
Service Validation: Employees for the bartender position must be 21 or over, and approved by a manager.