The cheese and ham toastie is the toastie with a layer that helps the seal instead of fighting it. The plain version lives or dies on the weld: buttered bread, cheese inside, pressed hot in an iron until the edges fuse and the centre flows. Sliding a slice of cooked ham into that build adds protein without adding a wet, loose filling that the seal has to contain, which is what sets it apart from the bean version. The ham is dry enough to sit inside the molten cheese without threatening the weld, so it reads as a richer, meatier toastie rather than a structurally riskier one.
The craft is letting the cheese still do the sealing with the ham in the way. The cheese has to be on both sides of the ham, not the ham against the bread, because the molten cheese is the gasket that welds the edges and binds the layers, and a slab of ham laid against the crumb would block that bond and leave a seam that splits on the cut. A strong Cheddar is chosen to melt smooth rather than split greasy under heat. The ham is folded rather than laid in a flat compressed sheet, so it keeps some loft and the bite has give instead of a dense plug, and it brings its own salt, which means the cheese can be a touch lighter than in a plain toastie. The iron runs hot to crisp and weld the buttered outside before the centre overheats.
The sealed-and-pressed frame holds the relatives. Cheese and beans trades the dry ham for a wet sauced filling and a harder seal problem; a slice of tomato or onion pushes water back in; the plain cheese toastie is this one with the ham removed. Each deserves its own article rather than being crowded in here.