One cannot hope to explain the meaning of dinner better than simply by quoting what The Oxford English Dictionary (OED) has to say..."The chief meal of the day, eaten originally and still by the majority of people about the middle of the day, but now, by the professional and fashionable classes, usually in the evening."
School meals are called school 'dinners' for the very simple reason that the vast majority of pupils who attended state schools when such meals were first provided were (still are?) the children of working-class parents. And the working folk of Britain do, by and large, still speak of the midday meal as 'dinner'. Obviously, therefore, that usage would be carried over from home to school.
It is not a case of anyone's meal-list being right or wrong. It is clear that this whole thing is almost entirely a matter of perceived class and - to some extent - geography.