You describe someone's idea that: "the word soup should be reserved
for warm liquids."
This seems an odd restriction. Suppose I buy a can of soup. Heat it. Consume some. Put some in the fridge for later. It starts cold, becomes hot, becomes warm, becomes cold again.
If it is soup only during the warm phase, what should I call it the rest of the time? Also, at what precise temperature does the cold 'stuff' transmogrify into 'soup'? It would certainly be possible to define the upper and lower temperatures but no-one is going to do that in real life.
Speaking personally, I enjoy tomato soup poured from a can into a mug and consumed cold - even with ice on a hot day. Should I change the name?
I would not call it tomato juice because that, to me, is the result of removing the liquid element from tomatoes with no additives (except perhaps some salt). By contrast, tomato soup often contains thickening agents as well as other extra ingredients - also it has been cooked at some point.
I believe the chief distinction between soup and juice is that the former has been cooked at some point even it is now cold. Cooking makes a chemical change that does not happen with juice.