I like to think of these models as the best of both worlds between handwritten rules and black-box models like DNNs. You get the control and user defined logic of handwritten rules through shape constraints, and the predictive power and flexibility of a black-box model through standard ML training techniques.

I think your description best fits the Calibrated Linear model, which is essentially a regression model with user defined logic/rules/constraints. But Calibrated Lattice and Calibrated Lattice Ensemble models are what we call "universal approximators" (just like DNNs), so you can approximate far more complex functions than simple regression.