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.
Way back in the day, when all we had was a space heater, my mom would sometimes crank the oven and then leave the door open. Just another space heater, but in the kitchen.
Often, before school, she would put a few slices of cheese on bread and then put them on a cookie sheet under the broiler (overhead heating element in an oven, in case that's not immediately clear). The cheese melts, the top of the bread toasts, and the underside steams a little bit.
You can eat them in slices, fold them over, or put two melty cheese bread slices together to make a sandwich, which was my favorite. It was one of the only things I knew how to cook for a long while, and I have fond memories of it.
Problem is that I never really knew what to call it. Mom called it cheese bread, but I never really thought that fit. Given my penchant for making sandwiches out of them, I eventually started calling them baked cheese sandwiches. I was hoping maybe cheese toasty would be the thing, but the search continues.
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!
We do something similar, but I'm super lazy. Basically, I do bread + mayo + shredded Mexican blend cheese, and microwave for 15 seconds or so until the cheese melts. If I'm not lazy, I'd butter the bottom of the broad and toast it on a pan w/ the lid on, so the bread gets a little toasty while the cheese melts.
We just call it "bread w/ cheese," because every other combination has a different meaning (cheese bread has cheese in the dough, cheesy bread has crispy cheese on the outside, grilled cheese has slices of [usually] cheddar). It's pretty good though, the cheese gets really melty, and it's really fast since the cheese is shredded.
It sounds pretty similar to what you make, and I think it's one of those "let's whip something up" dishes that don't get a proper name.
See around where i live, we call it a cheese toasty. We had class debates about if its a cheese toasty or a grilled cheese. Most people say a cheese toasty is when you take a pan on a stove or micowave. Then take 2 pieces of bread a sandwitch the cheese in the middle, then "toast it" sometimes including butter on the outside of the bread when using a pan. Grilled cheese is when you grill the sandwitch in the same way as toasting except its grilled.
Honestly its just kinda a silly little argument between people, i mean its bread with cheese! But its just fun to see what others call things. Very simular to coke/pop/soda debate.