OK. So let'ts take a controller with an explicit (I hope you agree) model, the Smith predictor. The controller as a whole has a model, but the subsystem C(z) (in the wiki example) has not (in your terms).

Or better yet, a Model Reference Adaptive Control. The system as a whole IS predictive, uses models, etc.. but the "core" controller subsystem does "not".

Then I'd argue that in the simple PID case, the engineer does the job of the Model/Adjusting Mechanism, and it's a fundamental part of the implementation process (you don't just buy a PID and install it without tuning it first!)

So, in every control systems there is a model. It's only that when the plant is "simple enough" and invariant in the long term the model/adjusting subsystem is implemented in wetware, and only used during install.

This is just arguing semantics, though.