Personally I'd lean towards grilled cheese with extra stuff in it for this sandwich. I feel like to be considered a melt it'd have to be something like a tuna melt where the cheese accompanies the tuna, and not the other way around. You think the classification is a question of volume proportions between cheese and whatever else is in it?
Nah with a melt, the other stuff is more important than the cheese. In a grilled cheese, the cheese is the star. Even in your link it says the cheese accompanies whatever else in a melt.