That's only one definition.
An equally valid one, not inconsistent with the original definition posted : Residential schools were government-sponsored schools established to introduce ALL children to the ethics and morals needed to function in modern, civilized societies and cultures.
There is no conflict with the presence or absence of actual religions, except ONE.
Which, to their eternal damnation, our Founding Fathers deemed too unnecessary or too obvious to define.
This may be a good time and place to express my strong opinion:
Teachers in ANY school must be deemed domestic terrorists and enemies of the State, and dismissed if they introduce "Personal politics" or psychological mental problems into their teaching.