Dinner is the largest meal of the day (not breakfast). Supper is food you eat after dinner. My mother grew up on a dairy farm in Kansas. Lots of farmers have their biggest meal at lunchtime. We had dinner at 6pm Mon-Sat. We had Sunday dinner about 2pm. Then at 6pm or so we had supper which was a get-your-own from what was put out. We did not sit at the table. It was the only time we could watch TV and eat.
I lived in New Zealand in the 1980's and found out that they also used supper as a "meal" after the largest meal. This was particularly true when attending a dance - supper was served around 11 pm. Their word for dinner was tea. So when someone invited you to tea you had to find out if it was for late afternoon (2:30-4:30) or later which was dinner.
I've always understood supper to be the evening meal, even if it's the main meal of the day, as it often is in North America. I imagine that's why the words supper and dinner are somewhat interchangeable there. Other places the distinction tends to be more clear. German avoids the issue completely, calling them early bit, midday meal, and evening meal respectively, though a more hearty brekky is often referred to as farmer's breakfast.
French has déjeuner (etymologically breakfast but chronologically lunch), petit déjeuner (little breakfast i.e. brekky), and dîner (dinner); plus the word souper exists, defined as a late supper. If I remember correctly (it's been decades since I've been to France), the main meal can be either midday or early evening.
In Spanish we have desayuno (breakfast, both etymologically and chron'ly), almuerzo (uncommon here in Mexico, the usual word is comida which also means full course meal), and cena (supper), which is almost always light. The main meal is almost always at midday, hence the supplanting of almuerzo with comida. Almuerzo really means lunch as opposed to full course midday meal. Merienda is a synomym for cena. (Folks down here continually tell me Spanish is difficult because it's riddled with synonyms, and I tell them I bet English is worse).
It appears that "school dinner" refers to the mid-day meal in the UK: Telegraph article.
Emmett
I've heard it explained different ways, but the way I settled it in my own mind is that dinner is the main or most complex meal of the day. In some families they have dinner at lunch; in others they have it at supper. In my family we had dinner for supper, except on Sunday when it was for lunch.
I grew up in the Midwest (WI) and for us "dinner" and "supper" were synonyms for the evening meal. Lunch was what you had at noon, usually a lighter meal. And breakfast, of course, was breakfast. I suspect there are variations on acceptable use in all regions of the country.