I separate my DD's food, and will be doing the same for DS when he starts school next year. They have one chiller bag, but 2 lunchboxes in that bag. So they know what food for which break.

I think I would appreciate that the teacher was doing something nice for you DS, not realising that he did in fact have enough food, but a bit concerned about the slushie. But I am guessing she couldn't go with him as she had to watch the other children. You have done the right thing talking to her and showing that you appreciate the kind gesture, but would prefer she didn't do it in future.

I am now feeling grateful for the policy at our school. If a child forgets lunch the school rings to make sure you are ok for them to provide something from the canteen (if the parents can't get to the school to drop food off). And the primary students ( we are at a prep- grade 12 school) aren't allowed to go to the canteen to choose, they need everything on a lunch order, and if an ice block is part of that, they get a token to take to the canteen after the first part of lunch.