As a kid in Sheffield I understood it as 'dinner' was your hot meal whenever you might have it, and tea was a lighter meal. So many kids had a hot meal at school (school dinner) and then went home for tea.
As a Norwegian in London, I still find it weird that my sons school serves "school dinner" around lunch-time. Even though the Norwegian word for dinner is "middag" which literally means "middle of the day" (though nobody uses it in that meaning any more, since it would be very confusing given that "middags-tider" "dinner-time" is pretty consistently late afternoon/early evening)
As an American in Norway, I still find it weird that warm lunches aren't a normal part of the school day. Then again, I remember school lunches. The kids aren't missing much.
The Swedes also have Frokost at breakfast. The word, via germanic roots, means "early food" (present day German früh = early) and used to denote a meal taken before noon (as made sense when farm workers rose with the sun), but not necessarily the first meal, which would be "morgenmad", or "morning food" in all three languages (and a much simpler meal, as nobody was up to prepare it, as opposed to the later meal). From there, as this meal became more redundant, the word drifted earlier in the day in Norwegian and Swedish to become breakfast, and later in Danish to become lunch.
The large meal taken in the middle of the day was "middagsmad", or "midday food" (still is in the older generations, especially rurally, in Denmark), while "middag" is generally the larger, evening meal today (so, same deal as "dinner" in English).
Interesting. Fenno-Swedish did away with "frukost", and the only morning meal is "morgonmål". Though I remember reading about someone having separate "morgonkaffe" and "frukost" as a child, and thinking it very strange...
From an Australian heritage, I would say that dinner and tea were, more or less, used interchangeably. Some families used one, some the other, but pretty much everyone understood both. Over time I get the feeling that dinner was the winner in that choice war, but that might just be an east coast/west coast thing.
My upbringing was middle-class, my dad taught for a time in Sheffield - tea just meant "evening meal" and was probably always a larger hot meal than was served at school dinners.
In South Wales they talk about "cooked dinner" which is a roasted meat meal, usually chicken for Sunday lunch, but can be any time of the week. Presumably that harks back to families having lots of uncooked meals (through poverty).
I should have clearer, I meant the roots of the phrase 'tea'.
That said I used to confuse some very northern friends by talking about dinner as my evening meal. They literally only thought of dinner as lunch and the evening meal was only called tea.
I think the roots were from when only the father of the house had a cooked evening meal, and everyone else had tea and sandwiches or similar. But that was distant past I believe.