Aussie here, I’ve always considered a cheese toasty to be a cheese sandwich that’s been toasted in a jaffle iron. A toasted cheese sandwich, however, would be more similar to a grilled cheese.
Not sure if this was just the nomenclature used by my family or is in line with what others call them, though.
Funnily enough, in Australia we call the broiler a grill, and what you’re describing is exactly what I thought a grilled cheese was. One of my favourite snacks when I was a kid was this, but with corn kernels under the cheese as well. Of course, being Australian, cheese and Vegemite was also a winning combination!