If we're going to dive straight in to the pedantry then: a panini, in English speaking countries is usually referring to a heated sandwich made from bread that is a roll (long rather than square, with an outer crust and sliced lengthways in half), usually some form of Italian bread in keeping with the Italian namesake. Panini's as far as I'm aware are filled with anything you want, but specifically are heated, usually (or exclusively?) in a press of some kind. Jaffles are like toasties, I'd personally call them a subset of toastie, heated in a specific type of press called a jaffle machine and made only with sliced, square, toast style bread as you'd likely get in a cheap, pre-sliced and packaged loaf. The type of press is important to qualify as a jaffle, as is the bread type and shape because these machines will only fit certain standardised bread types and needs to seal shut during heating. When you put a filled sandwich (with just about any filling combo but almost always with cheese), built with two, square, toasting slices, in to a jaffle machine the shape of the cavity in this machine forces a diagonal division between two opposing corners of the bread which also squashes the filling in to either of the two bread triangles formed on either side of this diagonal. The section of dividing line between the triangles compresses the two slices of bread together in that section, which gets particularly hot and forms a snappable, dark coloured ridge between the two halves of the jaffle. When your jaffle is done, it comes out as a single object with the two halves stuck together by the dividing line, but to eat, you typically apply pressure to each opposing half causing the brittle, dividing line to snap giving you two triangular halves of a sandwich with filling completely sealed inside.
You could perhaps say 'who calls a panini a toastie for $500?', because toasties have a much broader, looser definition like paninis. Even though the classic 'toastie' will more likely be similar to a jaffle, (though crucially not heated in a jaffle machine and thus not having the jaffle shape imposed upon it), it could actually be any bread and just about any filling (though almost always including cheese), much like a panini.
I really don't like jaffles and I have noticed a decline in their popularity as I've gotten older. They are a good idea in theory, but in practice, because the machine crimps the perimeter of the bread slices together and also the dividing line between the halves as well, you end up with burning hot filling and steam sealed and squashed inside of two bulged areas, one for each triangle. Those crimped edges and dividing line mean eating one involves a chore of biting through a lot of plain, unfilled, nearly burned toast before getting to all the filling which having been trapped inside is ridiculously hot and inevitably burns you. It also means that, the contents tends to get kind of steamed during cooking, making things quite flabby. Much prefer a toastie made in a sandwich press, which is basically a panini press minus the grill lines.