The jaffle is the sealed-edge toastie, and the seal is the entire distinction between it and an ordinary grilled one. Two slices of bread are buttered on the outside, filled with cheese or beans or tinned spaghetti, and clamped into a hinged stovetop iron with two rounded, scalloped jaws that press the bread together and cut a crimped weld right round the edge while it cooks over the hob. That crimped rim is the lead. A grilled toastie is sealed loosely if at all and tends to gape and shed its filling when it is cut; the jaffle is a closed pocket, welded shut on all four sides, so a hot and deliberately liquid centre is trapped completely and the first bite breaks a crust rather than opening a fold. The iron is not a faster grill, it is a different machine that makes a different object.
The craft is heat and load against that weld. The bite of crimped bread at the rim cooks denser and darker than the middle, so it is the part that tells you the iron has been on the heat long enough; pull it too early and the centre is cold inside a pale shell, leave it too long and the seam scorches. The butter on the outside is structural, conducting the heat into the bread and browning the crust the iron presses, and a dry face will toughen rather than crisp. The filling is kept loose and modest on purpose, because the appeal of a jaffle is the contrast of a near-molten, sometimes scalding interior against a hard sealed edge, and an overpacked one splits its weld and bleeds onto the iron.
The variations stay inside the sealed-iron frame and change only what is trapped. Cheese is the plain reference; baked beans or tinned spaghetti give the soft, hot, liquid centre the format is built around; a sweet jaffle traps jam or banana and turns the same weld to a pudding. Each deserves its own article rather than being crowded in here.